Abstract
In this paper, we relate Viterbo’s conjecture from symplectic geometry to Minkowski versions of worm problems which are inspired by the well-known Moser worm problem from geometry. For the special case of Lagrangian products this relation provides a connection to systolic Minkowski billiard inequalities and Mahler’s conjecture from convex geometry. Moreover, we use the above relation in order to transfer Viterbo’s conjecture to a conjecture for the longstanding open Wetzel problem which also can be expressed as a systolic Euclidean billiard inequality and for which we discuss an algorithmic approach in order to find a new lower bound. Finally, we point out that the above mentioned relation between Viterbo’s conjecture and Minkowski worm problems has a structural similarity to the known relationship between Bellmann’s lost-in-a-forest problem and the original Moser worm problem.
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1 Introduction and main results
Worm problems have a long history. The earliest known problem of this type was posed by Moser in [44] (see also [45]) more than 50 years ago:
Moser’s worm problem: Find a/the (convex) set of least area that contains a congruent copy of each arc in the plane of lenth one.
Here, the unit arcs are sometimes called worms, while the problem has been phrased in many different ways in the literature: the architect’s version (find the smallest comfortable living quarters for a unit worm), the humanitarian version (find the shape of the most efficient worm blanket), the sadistic version (find the shape of the best mallet head), and so on (see [53]). So far, despite a lot of research, only partial results are known, including the existence of such a minimum cover in the convex case (probably the first time proven in [39]), but its shape and area remain unknown. The best bounds presently known for its area \(\mu \) are:Footnote 1
(see [34] for the lower and [50] for the upper bound).
Worm problems can be formulated in considerable generality (see [53]):
Given a collection \({\mathcal {F}}\) of n-dimensional figures F and a transitive group \({\mathcal {M}}\) of motions m on \(\mathbb {R}^n\), find minimal convex target sets \(K\subset \mathbb {R}^n\)–minimal in the sense of having least volume, surface volume, or whatever–so that for each \(F\in {\mathcal {F}}\) there is a motion \(m \in {\mathcal {M}}\) with
$$\begin{aligned} m(F)\subseteq K. \end{aligned}$$
The existence of solutions to this problem can be guaranteed under certain natural hypotheses by fundamental compactness results like the Blaschke selection theorem (see [10, Sect. 18] for Blaschke’s selection theorem and [33, 39] for its application; see also Theorem 3.8 and its application in Propositions 3.9, 3.13, 3.19, and 3.20).
When the problem does not permit an arc to be replaced by its mirror image, then it is appropriate to consider the subgroup of orientation preserving motions. For other problems, e.g., Moser’s original worm problem, orientation reversing motions are permitted. Many problems whose motion group is the group of translations have been studied in the literature (see [8, 13, 52]).
In order to formulate the specific worm problem which is of main interest for our study, we introduce the following definition: Let \(T\subset \mathbb {R}^n\) be a convex body, i.e., a compact convex set in \(\mathbb {R}^n\) with nonempty interior, and \(T^\circ \) its polar. Using the Minkowski functional
with respect to T’s polar \(T^\circ \), we define the \(\ell _T\)-length of a closed \(H^1([0,\widetilde{T}],\mathbb {R}^n)\)-curveFootnote 2\(\dot{q}\) (from now on, for the sake of simplicity, every closed curve is assumed to fulfill this Sobolev property), \(\widetilde{T}\geqslant 0\), by
The worm problem which is of main interest for our study we call the Minkowski worm problem. Referring to the above general worm problem formulation, for this for convex body \(T\subset \mathbb {R}^n\), we consider \({\mathcal {F}}={\mathcal {F}}(T,\alpha )\) as the set of closed curves of \(\ell _T\)-length \(\alpha > 0\), \({\mathcal {M}}\) as the group of translations and the minimization in the sense of having minimal volume:
Minkowski worm problem: Let \(T\subset \mathbb {R}^n\) be a convex body. Find the volume-minimizing convex bodies \(K\subset \mathbb {R}^n\) that contain a translate of every closed curve of \(\ell _T\)-length \(\alpha \).
So, in contrast to Moser’s worm problem, we consider general dimension (instead of just dimension two), length-measuring with Minkowski functionals with respect to arbitrary convex bodies (instead of with respect to the Euclidean unit ball), closed curves (instead of not necessarily closed arcs), and translations (instead of congruence transformations). In other words and introducing a notation which will be useful throughout this paper: Let \(cc(\mathbb {R}^n)\) be the set of closed curves in \(\mathbb {R}^n\). Find the minimizersFootnote 3 of
where for convex body \(T\subset \mathbb {R}^n\) and \(\alpha > 0\), we define
with
and
where, for the sake of simplicity, we, in general, identify q with its image.
The only Minkowski worm problem that has been investigated so far is the case when the dimension is 2, T is the Euclidean unit ball in \(\mathbb {R}^2\), and, without loss of generality, \(\alpha =1\) (one could say: the two-dimensional Euclidean worm problem). It is known as:
Wetzel’s problem: Find the area-minimizing convex bodies \(K\subset \mathbb {R}^2\) that contain a translate of every closed curve of Euclidean length 1.
So far, the minimal area for this problem is not known, but the best bounds presently known for the minimum are 0.15544 as lower (see [52], where an argument from [47] is used) and 0.16526 as upper bound (see [8]; note that in [52] it was claimed incorrectly an upper bound of 0.159). In comparison to that: The areas of the obvious covers of constant width, the ball of radius 1/4 and the Reuleaux triangle of width 1/2, are 0.19635 and 0.17619, respectively. Since, by the Blaschke-Lebesgue theorem, the Reuleaux triangle is the area-minimizing set of constant width (see [9, 40]; see [27] for a direct proof by analyzing the underlying variational problem), we can conclude that a minimizer for Wetzel’s problem is not of constant width. We refer to Fig. 1 for three examples whose areas are approaching (not achieving) the minimum (clearly, the middle and right convex bodies are not of constant width).
Although we derive some results, the primary goal of our study will not be to solve these Minkowski worm problems, rather to relate them to Viterbo’s conjecture from symplectic geometry (see [49]) which for convex bodies \(C\subset \mathbb {R}^{2n}\) reads
For that, we recall that the EHZ-capacity of a convex body \(C\subset \mathbb {R}^{2n}\) can be definedFootnote 4 by
where a closed characteristic on \(\partial C\) is an absolutely continuous loop in \(\mathbb {R}^{2n}\) satisfying
where
\(\widetilde{T}\) is the period of the loop and by \(\mathbb {A}\) we denote its action defined by
The first main result of this paper addresses the special case of Lagrangian products
where K and T are convex bodies in \(\mathbb {R}^n\).Footnote 5 We denote by \({\mathcal {C}}(\mathbb {R}^n)\) the set of convex bodies in \(\mathbb {R}^n\).
Theorem 1.1
Viterbo’s conjecture for convex Lagrangian products \(K\times T\subset \mathbb {R}^n \times \mathbb {R}^n\)
is equivalent to the Minkowski worm problem
Additionally, equality cases \(K^*\times T^*\) of Viterbo’s conjecture satisfying
are composed of equality cases \((K^*,T^*)\) of (1). Conversely, equality cases \((K^*,T^*)\) of (1) form equality cases \(K^*\times T^*\) of Viterbo’s conjecture.
This yields the following corollary, which seems to be more suitable in order to approach Viterbo’s conjecture as an optimization problem (see Sect. 9).
Corollary 1.2
Viterbo’s conjecture for convex Lagrangian products \(K\times T\subset \mathbb {R}^n \times \mathbb {R}^n\)
is equivalent toFootnote 6
where the minimization runs for every \(q\in L_T(1)\) over all possible translations in \(\mathbb {R}^n\). Additionally, equality cases \(K^*\times T^*\) of Viterbo’s conjecture satisfying
are composed of equality cases \(T^*\) of (2) with
where \(a_q^*\) are the minimizers in (2). Conversely, equality cases \(T^*\) of (2) with \(K^*\) as in (3) form equality cases \(K^*\times T^*\) of Viterbo’s conjecture.
In analogy to Theorem 1.1, also Mahler’s conjecture from convex geometry (see [43]), i.e.,
where by \({\mathcal {C}}^{cs}(\mathbb {R}^n)\) we denote the set of all centrally symmetric convex bodies in \(\mathbb {R}^n\), can be expressed as a worm problem. As shown in [3], this is due to the fact that Mahler’s conjecture is a special case of Viterbo’s conjecture.
Theorem 1.3
Mahler’s conjecture for centrally symmetric convex bodies
is equivalent to the Minkowski worm problem
Additionally, equality cases \(T^*\) of Mahler’s conjecture (5) satisfying
are equality cases of (6). And conversely, equality cases \(T^*\) of (6) are equality cases of Mahler’s conjecture (5).
Furthermore, also systolic Minkowski billiard inequalities within the field of billiard dynamics can be related to worm problems.
In order to state this, let us recall some relevant notions from the theory of Minkowski billiards (see [36]): For convex bodies \(K,T\subset \mathbb {R}^n\), we say that a closed polygonal curveFootnote 7 with vertices \(q_1,...,q_m\), \(m\geqslant 2\), on the boundary of K is a closed weak (K, T)-Minkowski billiard trajectory if for every \(j\in \{1,...,m\}\), there is a K-supporting hyperplane \(H_j\) through \(q_j\) such that \(q_j\) minimizes
over all \(\bar{q}_j\in H_j\). We encode this closed (K, T)-Minkowski billiard trajectory by \((q_1,...,q_m)\). Furthermore, we say that a closed polygonal curve with vertices \(q_1,...,q_m\), \(m\geqslant 2\), on the boundary of K is a closed (strong) (K, T)-Minkowski billiard trajectory if there are points \(p_1,...,p_m\) on \(\partial T\) such that
is fulfilled for all \(j\in \{1,...,m\}\). We denote by \(M_{n+1}(K,T)\) the set of closed (K, T)-Minkowski billiard trajectories with at most \(n+1\) bouncing points.
Then, for convex body \(K\subset \mathbb {R}^n\), introducing \(F^{cp}(K)\) as the set of all closed polygonal curves in \(\mathbb {R}^n\) that cannot be translated into K’s interior \(\mathring{K}\), we have the following relations:
Theorem 1.4
Let \(T\subset \mathbb {R}^n\) be a convex body and \(\alpha ,c >0\). Then, the following statements are equivalent:
-
(1)
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(K)=c}\; \min _{q\in F^{cp}(K)}\ell _T(q) \leqslant \alpha , \quad K\in {\mathcal {C}}(\mathbb {R}^n), \end{aligned}$$
-
(2)
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(K)=c}\; c_{EHZ}(K\times T) \leqslant \alpha , \quad K\in {\mathcal {C}}(\mathbb {R}^n), \end{aligned}$$
-
(3)
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(K)=c}\;\; \min _{q \in M_{n+1}(K,T)} \ell _{T}(q) \leqslant \alpha , \quad K\in {\mathcal {C}}(\mathbb {R}^n), \end{aligned}$$
-
(4)
$$\begin{aligned} \min _{K\in A(T,\alpha )} {{\,\textrm{vol}\,}}(K)\geqslant c, \quad K\in {\mathcal {C}}(\mathbb {R}^n), \end{aligned}$$
-
(5)
$$\begin{aligned} \min _{a_q\in \mathbb {R}^n} {{\,\textrm{vol}\,}}\bigg (\textrm{conv}\bigg \{ \bigcup _{q\in L_T(1)}(q+a_q) \bigg \}\bigg ) \geqslant c, \quad K\in {\mathcal {C}}(\mathbb {R}^n). \end{aligned}$$
If T is additionally assumed to be strictly convex, then the following systolic weak Minkowski billiard inequality can be added to the above list of equivalent expressions:
-
(6)
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(K)=c}\;\; \min _{q \text { cl. weak }(K,T)\text {-Mink. bill. traj.}} \ell _{T}(q) \leqslant \alpha , \quad K\in {\mathcal {C}}(\mathbb {R}^n). \end{aligned}$$
Moreover, every equality case \((K^*,T^*)\) of any of the above inequalities is also an equality case of all the others.
Now, we turn our attention to the general Viterbo conjecture for convex bodies in \(\mathbb {R}^{2n}\). For that, we first introduce the following definitions: We denote by \({\mathcal {C}}^p\left( \mathbb {R}^{2n}\right) \) the set of convex polytopes in \(\mathbb {R}^{2n}\). For \(P\in {\mathcal {C}}^p\left( \mathbb {R}^{2n}\right) \), we denote by
the set of all closed polygonal curves \(q=(q_1,...,q_m)\) in \(F^{cp}(P)\) for which \(q_j\) and \(q_{j+1}\) are on neighbouring facets \(F_j\) and \(F_{j+1}\) of P such that there are \(\lambda _j,\mu _{j+1}\geqslant 0\) with
where \(x_j\) and \(x_{j+1}\) are arbitrarily chosen interior points of \(F_j\) and \(F_{j+1}\), respectively. Later, we will see that the existence of such closed polygonal curves is guaranteed.
Theorem 1.5
Viterbo’s conjecture for convex polytopes in \(\mathbb {R}^{2n}\)
is equivalent to the Minkowski worm problem
where we define
Additionally, \(P^*\) is an equality case of Viterbo’s conjecture for convex polytopes (7) satisfying
if and only if \(P^*\) is an equality case of (8).
When we look at the operator norm of the complex structure/symplectic matrix J with respect to a convex body \(C\subset \mathbb {R}^{2n}\) as map from
as it has been done in [2] and [23], i.e.,
then we derive the following theorem:
Theorem 1.6
Viterbo’s conjecture for convex bodies in \(\mathbb {R}^{2n}\)
is equivalent to
whereFootnote 8
Additionally, \(C^*\) is an equality case of Viterbo’s conjecture for convex bodies in \(\mathbb {R}^{2n}\) (9) satisfying
if and only if \(C^*\) is an equality case of (10).
Finally, we turn to Wetzel’s problem. For that, we keep the current state of things in mind:
Theorem 1.7
(Wetzel in [52], ’73; Bezdek and Connelly in [8], ’89) In dimension \(n=2\), we have
where we denote by \(B_1^2\) the Euclidean unit ball in \(\mathbb {R}^2\).
Then, as application of Theorem 1.1, we transfer Viterbo’s conjecture onto Wetzel’s problem. This results in the following conjecture:
Conjecture 1.8
We have
Applying [36, Theorem 3.12] and Theorem 1.4, we note that this conjecture can be equivalently expressed as systolic Euclidean billiard inequality:
Conjecture 1.9
We have
for \(K\in {\mathcal {C}}(\mathbb {R}^2)\).
We remark that, for the configuration \((K,B_1^2)\), due to the strict convexity of \(B_1^2\), the notions of weak and strong \((K,B_1^2)\)-Minkowski billiards coincide and are equal to the one of billiards in the Euclidean sense.
Although much work has been done around Wetzel’s problem and the systolic Euclidean billiard inequality, this shows that Viterbo’s conjecture is even unsolved for the “trivial” configuration
On the other hand, looking at these two problems from the symplectic point of view, can help us to conceptualize them from a very different point of view.
The Minkowski worm problems in Theorems 1.3, 1.5 and 1.6 seem to be very hard to solve (as it is expected from the perspective of Mahler’s/Viterbo’s conjecture). On the one hand, this is a consequence of the inner dependencies within
on the other hand, the right hand sides in (6), (8), and (10)
also contain dependencies and, beyond specific configurations, do not seem to be so accessible. Nevertheless, perhaps it turns out to be fruitful to investigate worm problems of the following structure a little bit more in detail: Find
Interestingly enough, from this perspective, Viterbo’s and Mahler’s conjecture are very similar in structure.
Motivated by a relationship between Moser’s worm problem and a version of Bellman’s lost-in-a-forest problem shown by Finch and Wetzel in [17], we further investigate whether it is possible also to relate Minkowski worm problems to versions of Bellman’s lost-in-a-forest problem. And indeed, it will turn out that the relationship established in [17] is somewhat similar to the relationship between Minkowski worm problems and Viterbo’s conjecture for convex Lagrangian products. However, before we will elaborate on this, we will give a short introduction to Bellman’s lost-in-a-forest problem and general escape problems of this type.
In 1955, Bellman stated in [5] the following research problem (see also [6] and [7]):
We are given a region R and a random point P within the region. Determine the paths which (a) minimize the expected time to reach the boundary, or (b) minimize the maximum time required to reach the boundary.
This problem can be phrased as:
A hiker is lost in a forest whose shape and dimensions are precisely known to him. What is the best path for him to follow to escape from the forest?
In other words: To solve the lost-in-a-forest problem one has to find the best escape path–the best in terms of minimizing the maximum or expected time required to escape the forest. A third interpretation of best has been given in [13]: Find the best escape path in terms of maximizing the probability of escape within a specified time period.
Bellman asked about two configurations in particular: on the one hand, the configuration in which the region is the infinite strip between two parallel lines a known distance apart, on the other hand, the configuration in which the region is a half-plane and the hiker’s distance from the boundary is known. For the case when best is understood in terms of the maximum time to escape, both of these two configurations have been studied: for the first configuration, the best path was found in [55] (’61), for the second, in [31] (’57) (where a complete and detailed proof was not published until it was done in [32] (’80); see [18] for an english translation). In each of these two cases, the shortest escape path is unique up to congruence. Apart from that, not much is known for other interpretations of best. We refer to [51] for a detailed survey on the different types, results, and some related material.
Finch and Wetzel studied in [17] the case in which the best escape path is the shortest. As already mentioned above, in this case, they could show a fundamental relation to Moser’s worm problem.
Before we further elaborate on this, it is worth mentioning to note that Williams in [54] has included lost-in-a-forest problems in his recent list “Million Buck Problems” of unsolved problems of high potential impact on mathematics. He justified the selection of these problems by mentioning that the techniques involved in their resolution will be worth at least one million dollars to mathematics.
Now, let’s consider the case studied by Finch and Wetzel and take it a little more rigorously. For that, let \(\gamma \) be a path in \(\mathbb {R}^2\), i.e., a continuous and rectifiable mapping of [0, 1] into \(\mathbb {R}^2\). Let \(\ell _{B_1^2}(\gamma )\) be its Euclidean length and \(\{\gamma \}\) its trace \(\gamma ([0,1])\). We call a forest a closed, convex region in the plane with nonempty interior. A path \(\gamma \) is an escape path for a forest K if a congruent copy of it meets the boundary \(\partial K\) no matter how it is placed with its initial point in K, i.e., for each point \(P\in K\) and each Euclidean motion (translation, rotation, reflection and combinations of them) \(\mu \) for which \(P=\mu (\gamma (0))\) the intersection \(\mu \left( \{\gamma \}\right) \cap \partial K\) is nonempty. Then, among all the escape paths for a forest K, there is at least one whose length is the shortest. The escape length \(\alpha \) of a forest K is the length of one of these shortest escape paths for K. Based on these notions, Finch and Wetzel proved the following:
Theorem 1.10
(Theorem 3 in [17]) Let \(K\subset \mathbb {R}^2\) be a convex body. The escape length \(\alpha ^*\) of K is the largest \(\alpha \) for which for every path \(\gamma \) with length \(\leqslant \alpha \), there is a Euclidean motion \(\mu \) such that K covers \(\mu \left( \{\gamma \}\right) \).
For Finch and Wetzel, this theorem established the connection to Moser’s worm problem. For that, we recall that in Moser’s worm problem one tries to find a/the convex set of least area that contains a congruent copy of each arc in the plane of a certain length. Clearly, the condition of having a certain length can be replaced by the condition of having a length which is bounded from above by that certain length.
Now, translated into our setting, we can derive a similar result. For that, we first have to define a version of a lost-in-a-forest problem which is compatible with the Minkowski worm problems discussed in the previous sections.
In order to indicate the connection to Minkowski worm problems in our setting, we will call the problem the Minkowski escape problem. We start by generalizing the problem to any dimension. So, we are considering higher dimensional “forests” which one aims to escape. We let \(K\subset \mathbb {R}^n\) be a convex body, measure lengths by \(\ell _T\), where \(T\subset \mathbb {R}^n\) is a convex body, and we call \(\gamma \) a closed Minkowski escape path for K if \(\gamma \) is a closed curve and for each point \(P\in K\) and each translation \(\mu \) for which \(P=\mu (\gamma (0))\) the intersection \(\mu \left( \{\gamma \}\right) \cap \partial K\) is nonempty. So, in contrast to considering not necessarily closed paths, allowing the motions to be Euclidean motions and measuring the lengths in the standard Euclidean sense in the escape problem of Finch and Wetzel, we only consider closed paths, translations and measure the lengths by the metric induced by the Minkowski functional with respect to the polar of T. Translating this problem into “our (mesocosmic) reality”—therefore, requiring \(n=2\) and Euclidean measurements, we get a slightly different problem (of course there are no limits to creativity) (see Fig. 2):
Two hikers walk in a forest. One of them gets injured and is in need of medical attention. The unharmed hiker would like to make the emergency call. Although he has his cell phone with him, there is only reception outside the forest. He has a map of the forest, i.e., the shape of the forest and its dimensions are known to him, and a compass to orient himself in terms of direction. Furthermore, he is able to measure the distance he has walked. However, he does not know exactly where in the forest he is. What’s the best way to get out of the forest, put off the emergency call, and then get back to the injured hiker?
The fact that in our story the unharmed hiker knows the shape of the forest and has a compass to orient himself in terms of direction is due to the fact that in our Minkowski escape problem, translations are the only allowed motions. The condition of coming back to the injured hiker is a consequence of our demand to consider only closed curves.
We can prove the analogue to Theorem 1.10:
Theorem 1.11
Let \(K,T\subset \mathbb {R}^n\) be convex bodies. Then, an/the \(\ell _T\)-minimizing closed Minkowski escape path for K has \(\ell _T\)-length \(\alpha ^*\) if and only if \(\alpha ^*\) is the largest \(\alpha \) for which
i.e., for which for every closed path \(\gamma \) of \(\ell _T\)-length \(\leqslant \alpha \), there is a translation \(\mu \) such that K covers \(\mu \left( \{\gamma \}\right) \).
Having in mind that Minkowski escape paths for a convex body \(K\subset \mathbb {R}^n\) can be understood as closed curves which cannot be translated into the interior of K, we can use the Minkowski billiard characterization of shortest closed polygonal curves that cannot be translated into the interior of K, in order to directly conclude the following corollary. Note for this line of argumentation that shortest closed curves that cannot be translated into the interior of K are in fact closed polygonal curves.
Corollary 1.12
Let \(K,T\subset \mathbb {R}^n\) be convex bodies, where T is additionally assumed to be strictly convex. An/The \(\ell _T\)-minimizing closed (K, T)-Minkowski billiard trajectory has \(\ell _T\)-length \(\alpha ^*\) if and only if \(\alpha ^*\) is the largest \(\alpha \) for which
So, the unharmed hiker in our story can conceptualize his problem by searching for length-minimizing closed Euclidean billiard trajectories.
In general, the problem of minimizing over Minkowski escape problems in the sense of varying the forest while maintaining their volume in order to find the forest with minimal escape length becomes the problem of solving systolic Minkowski billiard inequalities, or equivalently, the problem of proving/investigating Viterbo’s conjecture for Lagrangian products in \(\mathbb {R}^n\times \mathbb {R}^n\).
This means: If the hikers want to play it safe from the outset by choosing, among forests of equal area, the one where the time needed to help an injured hiker is minimized, then it is useful for them to be familiar with symplectic geometry or billiard dynamics. Of course, they could have paid attention from the beginning to where they entered the forest from and how they designed their path. Then they do not have to solve too difficult problems.
This paper is organized as follows: In Sect. 2, we start with some relevant preliminaries before, in Sect. 3, we derive properties of Minkowski worm problems and the fundamental results in order to prove Theorems 1.1, 1.3, 1.4, 1.5, and 1.6 and Corollary 1.2 in Sects. 4, 5, and 6. In Sect. 7, we prove that it is justified to transfer a special case of Viterbo’s conjecture into one for Wetzel’s problem which becomes Conjecture 1.8. In Sect. 8, we prove Theorem 1.10 as analogue to the relationship between Moser’s worm problem and Bellman’s lost-in-a-forest problem. Finally, in Sect. 9, we discuss a computational approach for improving lower bounds in Minkowski worm problems, especially lower bounds for Wetzel’s problem.
2 Preliminaries
We begin by collecting some results concerning the Fenchel–Legendre transform of a convex and continuous function \(H:\mathbb {R}^n\rightarrow \mathbb {R}\), which for \(x^*\in \mathbb {R}^n\) is defined by
Proposition 2.1
(Proposition II.1.8 in [14]) If \(H^*\) is the Fenchel–Legendre transform of a convex and continuous function \(H:\mathbb {R}^n\rightarrow \mathbb {R}\), then for \(x\in \mathbb {R}^n\) we have
The subdifferential of H in \(x\in \mathbb {R}^n\) is given by
Then, we get the Legendre recipocity formula:
Proposition 2.2
(Proposition II.1.15 in [14]) For a convex and continuous function \(H:\mathbb {R}^n\rightarrow \mathbb {R}\) the Legendre reciprocity formula is given by
where \(x,x^*\in \mathbb {R}^n\).
We state the generalized Euler identity:
Proposition 2.3
Let \(H:\mathbb {R}^n\rightarrow \mathbb {R}\) be a p-positively homogeneous, convex and continuous function of \(\mathbb {R}^n\). Then, for each \(x\in \mathbb {R}^n\) the following identity holds:
Proof
For each \(x\in \mathbb {R}^n\), since H is convex and continuous, we have
For each
Proposition 2.2 provides
and from Proposition 2.1, i.e.,
we get
Combining (11) and (12) we get
Now, we set
and recognize to have equality in (13) for \(\lambda \rightarrow 1\). Furthermore, we obtain by the p-homogeneity of H for \(\lambda \rightarrow 1\):
where we introduced the function
Because of
we get
\(\square \)
Noting that for convex body \(C\subset \mathbb {R}^n\)
is a 2-positively homogeneous, convex and continuous function, we derive the following properties:
Proposition 2.4
For convex body \(C\subset \mathbb {R}^n\) we have
Proof
For \(\xi \in \mathbb {R}^{n}\) we have
and therefore
\(\square \)
Proposition 2.5
Let \(C\subset \mathbb {R}^n\) be a convex body. If \(x^*\in \partial H_C(x)\) for \(x\in \mathbb {R}^n\), then
Proof
With Proposition 2.3 and the 2-homogeneity of \(H_{C^\circ }\) we can write
where
which together with Propositions 2.2 and 2.4 and
is equivalent to
Therefore, again using Proposition 2.3, we can conclude
In the following we show that
This would prove the claim.
Again, using Propositions 2.2 and 2.4, the fact
is equivalent to
All previous informations now can be summarized by the following two equations:
The difference yields
which implies
The conditions
imply, applying Proposition 2.3,
therefore
\(\square \)
The following proposition is the generalization of [36, Proposition 3.11] from closed polygonal curves to closed curves:
Proposition 2.6
Let \(T\subset \mathbb {R}^n\) be a convex body, \(q\in cc(\mathbb {R}^n)\) and \(\lambda >0\). Then, we have
Proof
From
(see [36, Proposition 2.3(iii)]) we conclude
and
\(\square \)
We continue by recalling [48, Theorem 1.1] which will be useful throughout this paper:
Theorem 2.7
Let \(K,T\subset \mathbb {R}^n\) be convex bodies. Then, we have
We note that in [48, Theorem 1.1] actually appear \(F^{cp}_{n+1}(K)\) and \(F^{cp}_{n+1}(T)\) instead of \(F^{cp}(K)\) and \(F^{cp}(T)\), respectively. However, for the purposes within this paper, we only need this more general formulation which is valid since there are no \(\ell _T\)/\(\ell _K\)-minimizing closed polygonal curves in \(F^{cp}(K)\)/\(F^{cp}(T)\) with more than \(n+1\) vertices and shorter \(\ell _T\)/\(\ell _K\)-length than the \(\ell _T\)/\(\ell _K\)-minimizing closed polygonal curves in \(F^{cp}_{n+1}(K)\)/\(F^{cp}_{n+1}(T)\) (see the proof of point (1) in the proof of [48, Theorem 2.2]).
We collect some invariance properties of Viterbo’s as well as of Mahler’s conjecture:
Proposition 2.8
Viterbo’s conjecture is invariant under translations.
Proof
Translations
are symplectomorphism because of
and therefore
Finally, we recall that Viterbo’s conjecture is invariant under symplectomorphisms, since symplectomorphisms in the above convex setting preserve the volume as well as the action and therefore the EHZ-capacity. \(\square \)
Proposition 2.9
Let \(C\subset \mathbb {R}^{2n}\) and \(K,T\subset \mathbb {R}^n\) be convex bodies. Then
for \(\lambda >0\), and
for \(\lambda ,\mu >0\). If
is an invertible linear transformation, then
Proof
We have
and
due to the 2-homogeneity of the action. Further,
and
due to Theorem 2.7 and [36, Proposition 3.11(ii) and (iv)] (see also Lemma 3.12).
Furthermore,
is a symplectomorphism, i.e.,
Indeed, for \(a,b\in \mathbb {R}^n\), we calculate
where we used the facts
Finally, we recall that, in the above convex setting, every symplectomorphism preserves the volume as well as the action and therefore the EHZ-capacity.
\(\square \)
Proposition 2.10
If \(T\subset \mathbb {R}^n\) is a centrally symmetric convex body and
an invertible linear transformation, then
Proof
Because of
and the volume preservation of
we have
\(\square \)
3 Properties of Minkowski worm problems
We begin by concluding some basic properties of the set
We note that all of the following properties can be easily extended to the case \(\alpha \geqslant 0\). Nevertheless, for the sake of simplicity and in order to avoid trivial case distinctions when it is not possible to divide by \(\alpha \), for the following we just treat the case \(\alpha >0\).
Proposition 3.1
Let \(T\subset \mathbb {R}^n\) be a convex body and \(\alpha ,\lambda ,\mu > 0\). Then we have
Proof
We have
Because of
(see Proposition 2.6) we conclude
which together with
implies
Again referring to Proposition 2.6 we conclude
and therefore
This implies
\(\square \)
Proposition 3.2
Let \(T\subset \mathbb {R}^n\) be a convex body and \(\alpha _1,\alpha _2 >0\). Then, we have
Proof
We find \(\mu >0\) such that
Then, using Proposition 3.1 we have
This implies
where the last equivalence follows from the following considerations: If we have (14) with \(\mu \geqslant 1\), then
means that
i.e.,
This implies
and therefore
The case \(\mu > 1\) follows by considering that in this case there can be find a convex body \(K^*\) with
Indeed, for
we define
Then, one has
and with (14)
with
it follows
by the definition of \(\widehat{K}\). \(\square \)
For convex body \(T\subset \mathbb {R}^n\) and \(\alpha > 0\) we define the set
where
Then, we have the following identity:
Proposition 3.3
Let \(T\subset \mathbb {R}^n\) be a convex body and \(\alpha > 0\). Then, we have
Proof
By definition it is clear that
Indeed, if
then this means
For \(\widetilde{\alpha }=\alpha \) it follows
and therefore
Let \(0<\widetilde{\alpha }\leqslant \alpha \). Then, it follows from Proposition 3.2 that
This implies
\(\square \)
Proposition 3.4
Let \(\alpha > 0\) and \(T_1,T_2\subset \mathbb {R}^n\) be two convex bodies. Then, we have
Proof
Let
If
then it follows from Proposition 3.3 that
Because of
as consequence of
it follows that
and therefore
With Proposition 3.3 this implies
Consequently, it follows
\(\square \)
Lemma 3.5
Let \(T\subset \mathbb {R}^n\) be a convex body and \(\alpha > 0\). Further, let \(K_1,K_2\subset \mathbb {R}^n\) be two convex bodies with
Then it holds:
Proof
Let
i.e.,
It obviously holds
Therefore
i.e.,
\(\square \)
For the next lemma we recall the following: If (M, d) is a metric space and P(M) the set of all nonempty compact subsets of M, then \((P(M),d_H)\) is a metric space, where by \(d_H\) we denote the Hausdorff metric \(d_H\) which for nonempty compact subsets X, Y of (M, d) is defined by
Then, \((cc(\mathbb {R}^n),d_H)\) is a metric subspace of the complete metric space \((P(\mathbb {R}^n),d_H)\) which is induced by the Euclidean space \((\mathbb {R}^n,|\cdot |)\). For convex body \(K\subset \mathbb {R}^n\) we consider
as metric subspaces of \((cc(\mathbb {R}^n),d_H)\). We have that
are complements of each other in \(cc(\mathbb {R}^n)\).
Lemma 3.6
Let \(K\subset \mathbb {R}^n\) be a convex body. Then, \((C(K),d_H)\) is a closed metric subspace of \((cc(\mathbb {R}^n),d_H)\).
Proof
Since C(K) is a subset of \(cc(\mathbb {R}^n)\), \((C(K),d_H)\) is a metric subspace of the metric space \((cc(\mathbb {R}^n),d_H)\). It remains to show that C(K) is a closed subset of \(cc(\mathbb {R}^n)\). For that let \((q_j)_{j\in \mathbb {N}}\) be a sequence of closed curves in C(K) \(d_H\)-converging to the closed curve \(q^*\). If
then \(q^*\) cannot be translated into K. Using the closedness of K in \(\mathbb {R}^n\) this means
Then, we can find a \(j_0\in \mathbb {N}\) such that
for all \(j\geqslant j_0\). But this implies
for all \(j\geqslant j_0\), i.e., \(p_j\) cannot be translated into K for all \(j\geqslant j_0\), a contradiction to
Therefore, it follows
and consequently, \((C(K),d_H)\) is a closed metric subspace of \((cc(\mathbb {R}^n),d_H)\). \(\square \)
Lemma 3.7
Let \(T\subset \mathbb {R}^n\) be a convex body and \((\alpha _k)_{k\in \mathbb {N}}\) an increasing sequence of positive real numbers converging to \(\alpha > 0\) for \(k\rightarrow \infty \). If
then also
Proof
Let
i.e.,
This means for all \(k\in \mathbb {N}\) that for all
holds
Let us assume that
i.e.,
This means that there is a \(q^*\in cc(\mathbb {R}^n)\) with
Due to Lemma \((C(K),d_H)\) is a closed metric subspace of \((cc(\mathbb {R}^n),d_H)\). Since \(F^{cc}(K)\setminus C(K)\) is the complement of C(K) in \(cc(\mathbb {R}^n)\) it follows that \(F^{cc}(K)\setminus C(K)\) is an open subset of the metric space \((cc(\mathbb {R}^n),d_H)\). Consequently there is a \(k_0\in \mathbb {N}\) sufficiently big such that
But with [36, Proposition 3.11(iv)] it is
i.e.,
which produces a contradiction between (15) and (16).
Therefore it follows
\(\square \)
The next proposition justifies to write “\(\min \)” instead of “\(\inf \)” within the Minkowski worm problem. The main ingredient of its proof will be Blaschke’s selection theorem (see [10, Sect. 18]).
Theorem 3.8
(Blaschke selection theorem) Let \((C_k)_{k\in \mathbb {N}}\) be a sequence of convex bodies in \(\mathbb {R}^n\) satisfying
for all \(k\in \mathbb {N}\). Then there is a subsequence \((C_{k_l})_{l\in \mathbb {N}}\) and a convex body C in \(\mathbb {R}^n\) such that \(C_{k_l}\) \(d_H\)-converges to C for \(l\rightarrow \infty \).
Proposition 3.9
Let T be a convex body and \(\alpha > 0\). Then we have
Proof
Let \((K_k)_{k\in \mathbb {N}}\) be a minimizing sequence of
Then, there is a \(k_0\in \mathbb {N}\) and a sufficiently big \(R>0\) such that
Indeed, if this is not the case, then there is a subsequence \(\left( K_{k_j}\right) _{j\in \mathbb {N}}\) such that
Guaranteeing
means that
The latter follows together with (18) and the convexity of \(K_{k_j}\) for all \(j\in \mathbb {N}\) from the fact that due to
there is no direction in which \(K_{k_j}\) can be shrunk. But (19) is not possible since \(\left( K_{k_j}\right) _{j\in \mathbb {N}}\) is a minimizing sequence of (17).
Applying Theorem 3.8, there is a subsequence \((K_{k_l})_{l\in \mathbb {N}}\) and a convex body \(K\subset \mathbb {R}^n\) such that \(K_{k_l}\) \(d_H\)-converges to K for \(l\rightarrow \infty \). It remains to show that
The fact that \(K_{k_l}\) \(d_H\)-converges to K for \(l\rightarrow \infty \) implies that for every \(\varepsilon >0\) there is an \(l_0\in \mathbb {N}\) such that
Then, with
it follows from the second inclusion in (20) together with Lemma 3.5 that
Applying Proposition 3.1 this means
We define the sequence
Then, \((\alpha _k)_{k\in \mathbb {N}}\) is an increasing sequence of positive numbers converging to \(\alpha \) for \(k\rightarrow \infty \) and together with the aboved mentioned (\(\varepsilon >0\) can be chosen arbitrarily) we have
Applying Lemma 3.7 it follows
\(\square \)
Proposition 3.10
Let \(T\subset \mathbb {R}^n\) be a convex body and \(\alpha ,\lambda ,\mu > 0\). Then we have
Proof
From
(see Proposition 3.1) it follows
and therefore
From
(see Proposition 3.1) it follows
and therefore
\(\square \)
Proposition 3.11
Let \(T\subset \mathbb {R}^n\) be a convex body and \(\alpha _1,\alpha _2 >0\). Then, we have
Proof
We find \(\mu >0\) such that
Then, we apply Proposition 3.10. \(\square \)
Now, for convex bodies \(K,T\subset \mathbb {R}^n\) we will turn our attention to the minimization problem
The existence of the minimum is guaranteed by Theorem 2.7.
Lemma 3.12
Let \(K,T\subset \mathbb {R}^n\) be convex bodies and \(\lambda >0\). Then
Proof
Similar to [36, Proposition 3.11(ii)] we have
and using [36, Proposition 3.11(iv)] therefore
\(\square \)
In the following for convex body \(T\subset \mathbb {R}^n\) and \(c>0\) we consider the minimax problemFootnote 9
The following proposition guarantees the existence of its maximum:
Proposition 3.13
Let \(T\subset \mathbb {R}^n\) be a convex body and \(c>0\). Then, we have
Proof
Let \((K_k)_{k\in \mathbb {N}}\) be a maximizing sequence of
Then, there is a \(k_0\in \mathbb {N}\) and an \(R>0\) such that
Indeed, if this is not the case, then there is a subsequence \(\left( K_{k_j}\right) _{j\in \mathbb {N}}\) such that
But this implies
This follows from the fact that for every \(j\in \mathbb {N}\) we can find a
with
The latter is a consequence of (23) and the constraint
i.e., due to the convexity of \(K_{k_j}\) for all \(j\in \mathbb {N}\) guaranteeing (25) there are directions from the origin in which \(K_{k_j}\) has to shrink for \(j\rightarrow \infty \) and which are suitable in order to construct convenient \(q_j\). But (24) is not possible since \(\left( K_{k_j}\right) _{j\in \mathbb {N}}\) is a maximizing sequence of (21).
Then, we apply Theorem 3.8 and find a subsequence \((K_{k_l})_{l\in \mathbb {N}}\) and a convex body \(K\subset \mathbb {R}^n\) such that \(K_{k_l}\) \(d_H\)-converges to K for \(l\rightarrow \infty \). It remains to prove that
but this is an immediate consequence of the \(d_H\)-continuity of the volume function. \(\square \)
Proposition 3.14
Let \(T\subset \mathbb {R}^n\) be a convex body. Then,
increases/decreases strictly if and only if this is the case for \(c> 0\).
Proof
We make use of the implication
for all \(c_1,c_2>0\).
Let us verify (26): We assume
and without loss of generality \(c_1 < c_2\). Let the pair
be a maximizer of the left side in (27), i.e.,
With
similar to [36, Proposition 3.11(ii)] we have
for
From
it follows together with Lemma 3.12 that
Since
we conclude
which is a contradiction to (27). Therefore, noting that the assumption \(c_1 > c_2\) would have led analogously to the same contradiction, it follows
We now prove the equivalence
for \(c_1,c_2>0\).
If
then from the first part of the proof it necessarily follows \(c_1 \ne c_2\). Let us assume \(c_1>c_2\). We further assume that the pair
is a maximizer of the right side in (29), i.e.,
We define
From
it follows together with Lemma 3.12 that
Since
we conclude
which is a contradiction to (29). Therefore, we conclude \(c_1<c_2\).
Conversely, let \(c_1<c_2\). From (26) we conclude
If the strict inequality “>” holds in (30), then we conclude from the above proven implication “\(\Rightarrow \)” in (28) that \(c_1 > c_2\), a contradiction. Therefore, it follows
\(\square \)
Proposition 3.15
Let \(K,T\subset \mathbb {R}^n\) be convex bodies with \(q^*\) as minimizer of
Then, it follows
Proof
Let \(q^*\) be a minimizer of
Then, it follows
Indeed, otherwise, if there is
i.e.,
then, due to the openess of
with respect to \(d_H\) (see Lemma ), there is a \(\lambda <1\) such that
Then, using [36, Proposition 3.11(iv)], we conclude
Because of the \(d_H\)-density of \(F^{cp}(K)\) in \(F^{cc}(K)\) and the \(d_H\)-continuity of \(\ell _T\) on \(F^{cc}(K)\) (see [36, Proposition 3.11(v)]–which is also valid for closed curves) then we can find a
with
a contradiction.
Finally, from
it follows
\(\square \)
Lemma 3.16
Let \(K\subset \mathbb {R}^n\) be a convex body and \(\lambda >1\). If
then it follows that
Proof
If we assume (31) but (32) does not hold. Then it follows
and due to \(\lambda >1\) therefore
But this is a contradiction to
Therefore, it follows (32). \(\square \)
Proposition 3.17
Let \(T\subset \mathbb {R}^n\) be a convex body and \(\alpha > 0\). If \(K^*\) is a minimizer of
then
Proof
If \(q^*\) is a minimizer of
then it follows from Proposition that
This means
Proposition 3.11 implies
If
then with Proposition [36, Proposition 3.11(iv)] there is \(\lambda >1\) such that
Together with
and Lemma 3.16 the fact
implies
therefore, there is no translate of \(K^*\) that covers \(\lambda q^*\). Consequently,
a contradiction to the fact that \(K^*\) is a minimizer of (33). Therefore, it follows that
\(\square \)
The idea which underlies the following theorem leads to the heart of this paper.
Theorem 3.18
Let \(T\subset \mathbb {R}^n\) be a convex body. If \(K^*\) is a minimizer of
for \(\alpha > 0\), then \(K^*\) is a maximizer of
for
with
Conversely, if \(K^*\) is a maximizer of (35) for \(c>0\), then \(K^*\) is a minimizer of (34) for
and with
Consequently, for \(\alpha , c>0\) we have the equivalence
and moreover
Proof
Let \(K^*\) be a minimizer of (34) for \(\alpha >0\). If \(K^*\) is not a maximizer of (35) for
then there is a convex body
and a
such that
where by \(q^*\) we denote a minimizer of
From Proposition it follows
and further from Proposotion 3.17 that
From (37), (38) and (39) together with Proposition 3.11 we conclude
a contradiction. Therefore, \(K^*\) is a maximizer of (35) for
Conversely, let \(K^*\) be a maximizer of (35) for \(c>0\) with
such that
Then, from Proposition it follows that
and consequently
If \(K^*\) is not a minimizer of (34) for
then there is a
with
Then, from Proposition 3.17 it follows that
This implies
which because of (40) is a contradiction to Proposition 3.14. We conclude that \(K^*\) is a minimizer of (34) for
From the before proven it clearly follows the equivalence
for \(\alpha ,c >0\). In order to prove (36) it remains to show
Let \(K^*\) be a minimizer of
where \(c>0\) is chosen such that
Then we know from the above reasoning that \(K^*\) is a maximizer of
with
From (41) and Proposition 3.14 it follows
Conversely, let \(K^*\) be a maximizer of
where \(\alpha >0\) is chosen such that
Then we know from the above reasoning that \(K^*\) is a minimizer of
and from Proposition 3.11 it follows
\(\square \)
Hereinafter we will deal with the following two minimax problemsFootnote 10: For \(\alpha ,d >0\) we will consider
and for \(c,d >0\) we will consider
It is indeed justified to write “\(\min \)” and “\(\max \)” respectively:
Proposition 3.19
Let \(\alpha ,d > 0\). Then we have
Proof
Let \((T_k)_{k\in \mathbb {N}}\) be a minimizing sequence of
Then there is an \(R>0\) and a \(k_0\in \mathbb {N}\) such that
Indeed, if this is not the case, then there is a subsequence \(\left( T_{k_j}\right) _{j\in \mathbb {N}}\) such that
This implies
The latter follows from the fact that—(43) together with the convexity of \(T_{k_j}\) for all \(j\in \mathbb {N}\) and the constraint
means that there are directions from the origin in which \(T_{k_j}\) has to shrink for \(j\rightarrow \infty \)–for every \(j\in \mathbb {N}\) we can find
(\(q_j\) can be constructed by using the aforementioned directions) for which
means
where by \(\widetilde{T}_j\) we denote the period of the closed curve \(q_j\), and for every convex body \(K_j\subset \mathbb {R}^n\) minimizing
means
But this is not possible since \((T_k)_{k\in \mathbb {N}}\) is a minimizing sequence of (42).
Then, we can apply Theorem 3.8: There is a subsequence \(\left( T_{k_l}\right) _{l\in \mathbb {N}}\) and a convex body \(T\subset \mathbb {R}^n\) such that \(T_{k_l}\) \(d_H\)-converges to T for \(l\rightarrow \infty \). We clearly have
Therefore, T is a minimizer of (42). \(\square \)
Proposition 3.20
Let \(c,d >0\). Then we have
Proof
Let \(\alpha >0\) and let us consider the minimax problem
Let the pair
be a minimizer of (45), i.e., it is
By Theorem 3.18\(K^*\) is a maximizer of
with
Then, due to
we clearly have
If this is a strict inequality, then there is a pair of convex bodies
such that
Then, by Theorem 3.18\(K^{**}\) is a minimizer of
with
Now, \(\widetilde{\alpha } >\alpha \) together with Proposition 3.11 implies
a contradiction. Therefore, it follows that the inequality in (46) is in fact an equality, i.e.,
This means that the pair \((K^*,T^*)\) is a maximizer of
Since it is sufficient to prove the claim (44) for one \(c>0\), we are done. \(\square \)
Theorem 3.21
If the pair \((K^*,T^*)\) is a minimizer of
for \(\alpha ,d >0\), then \((K^*,T^*)\) is a maximizer of
for
with
Conversely, if the pair \((K^*,T^*)\) is a maximizer of (48) for \(c,d>0\), then \((K^*,T^*)\) is a minimizer of (47) for
with
Consequently, for \(\alpha ,c,d>0\) we have the equivalence
and moreover
Proof
Let the pair \((K^*,T^*)\) be a minimizer of (47) for \(\alpha ,d>0\), i.e., it is
such that
Then, in the proof of Proposition 3.20 we have seen that \((K^*,T^*)\) is a maximizer of (48) for
Conversely, let the pair \((K^*,T^*)\) be a maximizer of (48) for \(c,d>0\), i.e., \(K^*,T^*\subset \mathbb {R}^n\) are convex bodies of volume c and d, respectively, such that
By Theorem 3.18\(K^*\) minimizes
with
Then, we clearly have
If this is a strict inequality, then there is a pair \((K^{**},T^{**})\) with
where
and \(T^{**}\subset \mathbb {R}^n\) is a convex body of volume d. Then, by Theorem 3.18\(K^{**}\) is a maximizer of
with
Now, \(\widetilde{c}<c\) together with Proposition 3.14 implies
a contradiction. Therefore,
i.e., the pair \((K^*,T^*)\) is a minimizer of (47).
From the before proven it clearly follows the equivalence
for \(\alpha , c, d >0\).
In order to prove (49) it is sufficient to show
Let the pair \((K^*,T^*)\) be a minimizer of
where \(c>0\) is chosen such that
From above reasoning we know that \((K^*,T^*)\) maximizes (48) (for c replaced by \(\widetilde{c}\)), i.e., \(K^*,T^*\subset \mathbb {R}^n\) are convex bodies of volume \(\widetilde{c}\) and d, respectively, such that
Now, \(c<\widetilde{c}\) together with Proposition 3.14 implies
Conversely, let \((K^*,T^*)\) be a maximizer of
i.e., \(K^*,T^*\subset \mathbb {R}^n\) are convex bodies of volume c and d, respectively, where \(\alpha >0\) is chosen such that
Then we know from above reasoning that \((K^*,T^*)\) minimizes (47) (for \(\alpha \) replaced by \(\widetilde{\alpha }\)), i.e.,
Now, \(\alpha > \widetilde{\alpha }\) together with Proposition 3.11 implies
\(\square \)
4 Proofs of Theorems 1.1, 1.3, 1.4 and Corollary 1.2
In the following, we mainly make use of Theorems 3.18 and 3.21. However, we begin by rewriting Viterbo’s conjecture for convex Lagrangian products:
Proposition 4.1
Viterbo’s conjecture for convex Lagrangian products \(K\times T \subset \mathbb {R}^n\times \mathbb {R}^n\)
is equivalent to
Proof
Using Proposition 2.9, Viterbo’s conjecture for convex Lagrangian products is equivalent to
By Theorem 2.7, this is equivalent to
\(\square \)
Now, we can prove Theorems 1.1, 1.3, 1.4 and Corollary 1.2 which we will recall for the sake of overview, respectively.
Theorem
(Theorem 1.1) Viterbo’s conjecture for convex Lagrangian products \(K\times T\subset \mathbb {R}^n \times \mathbb {R}^n\)
is equivalent to the Minkowski worm problem
Additionally, equality cases \(K^*\times T^*\) of Viterbo’s conjecture satisfying
are composed of equality cases \((K^*,T^*)\) of (50). Conversely, equality cases \((K^*,T^*)\) of (50) form equality cases \(K^*\times T^*\) of Viterbo’s conjecture.
Proof
Using Proposition 4.1, Viterbo’s conjecture for convex Lagrangian products is equivalent to
After applying Theorem 3.21, it is further equivalent to
Using Proposition 3.10, this can be written as
By similar reasoning, Theorem 3.21 also guarantees the equivalence of the equality case of Viterbo’s conjecture for convex Lagrangian products \(K\times T\subset \mathbb {R}^n\times \mathbb {R}^n\)
i.e.,
and
Moreover, Theorem 3.21 guarantees the following: If \(K^*\times T^*\) is a solution of (51) satisfying
(note that, applying Proposition 2.9, the property of being a solution of (51) is invariant under scaling), then the pair \((K^*,T^*)\) is a solution of (52). And conversely, if the pair \((K^*,T^*)\) is a solution of (52), then \(K^*\times T^*\) is a solution of (51). \(\square \)
Corollary
(Corollary 1.2) Viterbo’s conjecture for convex Lagrangian products \(K\times T\subset \mathbb {R}^n \times \mathbb {R}^n\)
is equivalent toFootnote 11
where the minimization runs for every \(q\in L_T(1)\) over all possible translations in \(\mathbb {R}^n\). Additionally, equality cases \(K^*\times T^*\) of Viterbo’s conjecture satisfying
are composed of equality cases \(T^*\) of (54) with
where \(a_q^*\) are the minimizers in (54). Conversely, equality cases \(T^*\) of (54) with \(K^*\) as in (55) form equality cases \(K^*\times T^*\) of Viterbo’s conjecture.
Proof
In view of the proof of Theorem 1.1, for convex bodies \(K,T\subset \mathbb {R}^n\), it is sufficient to prove the following equality:
But this follows from the following gradually observation: First, we notice that the volume-minimizing convex cover for a set of closed curves is, equivalently, the volume-minimizing convex hull of these closed curves. So, if we ask for lower bounds of
we note that for \(q_1,...,q_k\in L_T(1)\), we have
This estimate can be further improved by
so that eventually we get
where the minimum on the left runs for every \(q\in L_T(1)\) over all possible translations in \(\mathbb {R}^n\). Minimizing this equation over all convex bodies \(T\subset \mathbb {R}^n\) of volume 1, we get (56). \(\square \)
Theorem
(Theorem 1.3) Mahler’s conjecture for centrally symmetric convex bodies
is equivalent to the Minkowski worm problem
Additionally, equality cases \(T^*\) of Mahler’s conjecture (57) satisfying
are equality cases of (58). And conversely, equality cases \(T^*\) in (58) are equality cases of Mahler’s conjecture (57).
Proof
Because of
for all centrally symmetric convex bodies \(T\subset \mathbb {R}^n\) (see [3]), Mahler’s conjecture for centrally symmetric convex bodies is equivalent to
Fixing
which is without loss of generality due to Proposition 2.10, and using Theorem 2.7, (59) is equivalent to
This can be written as
which, by Theorem 3.18, is equivalent to
Applying Proposition 3.10, we finally conclude that Mahler’s conjecture for centrally symmetric convex bodies is equivalent to
By similar reasoning, Theorem 3.18 also guarantees the equivalence of the equality case of Mahler’s conjecture for centrally symmetric convex bodies \(T\subset \mathbb {R}^n\)
i.e.,
and
Moreover, Theorem 3.18 guarantees the following: If \(T^*\) is a solution of (60) satisfying
(note that, applying Proposition 2.10, the property of being a solution of (60) is invariant under scaling), then it is a solution of (61). And conversely, if \(T^*\) is a solution of (61), then it is also a solution of (60). \(\square \)
Theorem
(Theorem 1.4) Let \(T\subset \mathbb {R}^n\) be a convex body and \(\alpha ,c >0\). Then, the following statements are equivalent:
-
(1)
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(K)=c}\; \min _{q\in F^{cp}(K)}\ell _T(q) \leqslant \alpha , \quad K\in {\mathcal {C}}(\mathbb {R}^n), \end{aligned}$$
-
(2)
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(K)=c}\; c_{EHZ}(K\times T) \leqslant \alpha , \quad K\in {\mathcal {C}}(\mathbb {R}^n), \end{aligned}$$
-
(3)
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(K)=c}\;\; \min _{q \in M_{n+1}(K,T)} \ell _{T}(q) \leqslant \alpha , \quad K\in {\mathcal {C}}(\mathbb {R}^n), \end{aligned}$$
-
(4)
$$\begin{aligned} \min _{K\in A(T,\alpha )} {{\,\textrm{vol}\,}}(K)\geqslant c, \quad K\in {\mathcal {C}}(\mathbb {R}^n), \end{aligned}$$
-
(5)
$$\begin{aligned} \min _{a_q\in \mathbb {R}^n} {{\,\textrm{vol}\,}}\bigg (\textrm{conv}\bigg \{ \bigcup _{q\in L_T(1)}(q+a_q) \bigg \}\bigg ) \geqslant c, \quad K\in {\mathcal {C}}(\mathbb {R}^n). \end{aligned}$$
If T is additionally assumed to be strictly convex, then the following systolic weak Minkowski billiard inequality can be added to the above list of equivalent expressions:
-
(6)
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(K)=c}\;\; \min _{q \text { cl. weak }(K,T)\text {-Mink. bill. traj.}} \ell _{T}(q) \leqslant \alpha , \quad K\in {\mathcal {C}}(\mathbb {R}^n). \end{aligned}$$
Moreover, every equality case \((K^*,T^*)\) of any of the above inequalities is also an equality case of all the others.
Proof
The equivalence of (1), (2), and (3) follows from Theorem 2.7. The equivalence of (1) and (4) follows from Theorem 3.18. The equivalence of (4) and (5) can be concluded as within the proof of Corollary 1.2. For the case of strictly convex \(T\subset \mathbb {R}^n\), the equivalence of (1) and (6) follows from [36, Theorem 1.3].
The addition that every equality case \((K^*,T^*)\) of any of the inequalities is also an equality case of all the others is guaranteed by Theorem 3.18. \(\square \)
5 Proof of Theorem 1.5
We start by recalling Theorem 1.5:
Theorem
(Theorem 1.5) Viterbo’s conjecture for convex polytopes in \(\mathbb {R}^{2n}\)
is equivalent to the Minkowski worm problem
where we define
Additionally, \(P^*\) is an equality case of Viterbo’s conjecture for convex polytopes (62) satisfying
if and only if \(P^*\) is an equality case of (63).
Now, we recall a sligthly rephrased version of the main result of Haim-Kislev in [26]:
Theorem 5.1
Let \(P\subset \mathbb {R}^{2n}\) be a convex polytope. Then, there is an action-minimizing closed characteristic x on \(\partial P\) which is a closed polygonal curve consisting of finitely many segments
given by
while \(x_j\in \mathring{F}_j\), \(F_j\) is a facet of P and x visits every facet \(F_j\) at most once.
For the proof of Theorem 1.5, we need the following theorem:
Theorem 5.2
If \(P\subset \mathbb {R}^{2n}\) is a convex polytope, then we have
with
If we consider \(P\times \frac{1}{2}JP\) as a Lagrangian product (in the light of Footnote 8 within Theorem 1.6), then the combination of Theorem 2.7 and Theorem 5.2 implies the following relationship between the EHZ-capacity of P and the EHZ-capacity of the Lagrangian product \(P\times \frac{1}{2}JP\):
For the proof of Theorem 5.2, we need the following proposition. We remark that in the proof of Theorem 5.2, we need it only in the case of action-minimizing closed characteristics on the boundary of a polytope. However, we will state it in full generality which has relevance beyond its use in the proof of Theorem 5.2 (which we will briefly address below).
Proposition 5.3
Let \(C\subset \mathbb {R}^{2n}\) be a convex body. Let x be any closed characteristic on \(\partial C\). Then, the action of x equals its \(\ell _{\frac{JC}{2}}\)-length:
Proposition 5.3 implies a noteworthy connection between closed characteristics and closed Finsler geodesics: Every closed characteristic on \(\partial C\) can be interpreted as a closed Finsler geodesic with respect to the Finsler metric determined by \(\mu _{2JC^\circ }\) and which is parametrized by arc length. This raises a number of questions; for example, which closed Finsler geodesics are closed characteristics (we note that there are more closed geodesics than those which, by the least action principle and Proposition 5.3, can be associated to closed characteristics) and the length-minimizing closed Finsler geodesics of which class are of this kind. Following this line of thought, would lead to the question whether it is possible to deduce Viterbo’s conjecture from systolic inequalities for certain closed Finsler geodesics. However, we leave these questions for further research.
Proof of Proposition 5.3
By
we conclude
where we used the facts
and
(see [36, Proposition 2.3(iii)]). From Proposition 2.5, we therefore conclude
and consequently
Considering
(see [36, Proposition 2.1]), we obtain
where the last equality follows from
which by Proposition 2.3 and the 2-homogeneity of \(H_C\) implies
\(\square \)
Then, we come to the proof of Theorem 5.2:
Proof of Theorem 5.2
The idea behind the proof is to associate action-minimizing closed characteristics on \(\partial P\) in the sense of Theorem 5.1 with \(\ell _{\frac{1}{2}JP}\)-minimizing closed \((P,\frac{JP}{2})\)-Minkowski billiard trajectories.
Let x be an action-minimizing closed characteristic on \(\partial P\) in the sense of Theorem 5.1. Let us assume x is moving on the facets of P according to the order
while the linear flow on every facet is given by the J-rotated normal vector at the interior of this facet. Out of every trajectory segment
we choose one point \(q_j\) arbitrarily (on the whole requiring \(q_i\ne q_j\) for \(i\ne j\)) and connect these points by straight lines (by maintaining the order of the corresponding facets) constructing a closed polygonal curve
within P which has its vertices on \(\partial P\). From Lemma 5.4 (which we provide subsequently), we derive
since the trajectory segment of x between the two consecutive points \(q_j\) and \(q_{j+1}\)–let us call it \(\textrm{orb}(x)_{q_j\rightarrow q_{j+1}}\)–together with the line from \(q_j\) to \(q_{j+1}\) (as trajectory segment of q)–let us call it \([q_j,q_{j+1}]\)–builds a triangle with the property that
We therefore conclude from Proposition 5.3 that
Because of the arbitrariness of the choice of \(q_j\) within \(\textrm{orb}(x)\cap \mathring{F}_j\), we can assign infinitely many different closed polygonal curves of the above kind to one action-minimizing closed characteristic fulfilling the demanded conditions.
Each of these closed polygonal curves q is a closed \((P\times \frac{1}{2}JP)\)-Minkowski billiard trajectory: This follows from the fact that q fulfills
for the closed polygonal curve \(p=(p_1,...,p_m)\) in \(\frac{1}{2}JP\) with
given as the intersection point
for all \(j\in \{2,...,m+1\}\).
Indeed, from the definition of p, it follows
since by construction, \(p_{j+1}-p_j\) is a multiple of the outer normal vector at P in \(q_j\) rotated by twofold multiplication with J (\(J^2=-\mathbbm {1}\) produces the minus sign in (64)). Since, by construction,
is in the normal cone at P in the intersection point
roation by \(\frac{1}{2}J\) then implies that \(q_j-q_{j-1}\) is in the normal cone at \(\frac{1}{2}JP\) in \(p_{j-1}\). This implies
From [36, Proposition 3.9], it follows that q cannot be translated into \(\mathring{P}\), i.e.,
From the construction of q, we moreover know
where we recall that \(F^{cp}_*(P)\) as subset of \(F^{cp}(P)\) was defined as the set of all closed polygonal curves \(q=(q_1,...,q_m)\) in \(F^{cp}(P)\) for which \(q_j\) and \(q_{j+1}\) are on neighbouring facets \(F_j\) and \(F_{j+1}\) of P such that there are \(\lambda _j,\mu _{j+1}\geqslant 0\) with
where \(x_j\) and \(x_{j+1}\) are arbitrarily chosen interior points of \(F_j\) and \(F_{j+1}\), respectively.
Because of (65), we have
Since, by definition and the above considerations, every closed polygonal curve in \(F^{cp}_*(P)\) is associated with a closed characteristic on \(\partial P\), where the \(\ell _{\frac{JP}{2}}\)-length of the former coincides with the action of the latter, and x (to which q is associated) was chosen to be action-minimizing, we actually have
Altogether, this implies
for
\(\square \)
Lemma 5.4
Let \(P\subset \mathbb {R}^{2n}\) be a convex polytope. If
where \(F_i\) and \(F_j\) are neighbouring facets of P with \(x_i\in \mathring{F}_i\) and \(x_j\in \mathring{F}_j\), then
Proof
We first notice that
are neighbouring vertices of \(P^\circ \), i.e.,
Indeed, from the fact that \(\nabla H_P(x_i)\) and \(\nabla H_P(x_j)\) are elements of the one dimensional normal cone at \(\mathring{F}_i\) and \(\mathring{F}_j\), we conclude by the properties of the polar of convex polytopes (see [21, Chapter 3.3]) that they point into the direction of two neigbouring vertices of \(P^\circ \). Using Proposition 2.5, we calculate
and
and conclude that \(\nabla H_P(x_i)\) and \(\nabla H_P(x_j)\) actually are these two neighbouring vertices of \(P^\circ \).
Using for convex body \(C\subset \mathbb {R}^{2n}\) and \(\lambda > 0\) the properties
(see [36, Proposition 2.3(iii)]), we derive
where in \((\star )\) we used that, by the choice of \(x_i\) and \(x_j\) and the properties of polar bodies, \(\nabla H_P(x_i)\) and \(\nabla H_P(x_j)\) are neighbouring vertices of \(P^\circ \) and, therefore, in \((\star )\), the initial term can be splitted linearly. \(\square \)
Proof of Theorem 1.5
Viterbo’s conjecture for convex polytopes in \(\mathbb {R}^{2n}\) can be written as
which by Theorem 5.2, is equivalent to
By referring to Proposition 2.9, we can assume
without loss of generality and get
which by Theorem 2.7, is equivalent to
By Theorem 3.18, this is equivalent to
and after applying Proposition 3.10, to
By similar reasoning, Theorem 3.18 also guarantees the equivalence of
and
Moreover, Theorem 3.18 guarantees the following: \(P^*\) is a solution of (66) if and only if \(P^*\) is a solution of (67). \(\square \)
6 Proof of Theorem 1.6
We start by recalling Theorem 1.6:
Theorem
(Theorem 1.6) Viterbo’s conjecture for convex bodies in \(\mathbb {R}^{2n}\)
is equivalent to the Minkowski worm problem
where
Additionally, \(C^*\) is an equality case of Viterbo’s conjecture for convex bodies in \(\mathbb {R}^{2n}\) (68) satisfying
if and only if \(C^*\) is an equality case of (69).
In order to prove Theorem 1.6, we need the following propositon:
Proposition 6.1
Let \(C\subset \mathbb {R}^{2n}\) be a convex body and x a closed characteristic on \(\partial C\). Then, x cannot be translated into \(\mathring{C}\).
Proof
Let us assume that x can be translated into \(\mathring{C}\). Let \(\widetilde{T}>0\) be the period of x. Because of
there is a vector-valued function \(n_C\) on \(\partial C\) such that
with
for all \(t\in [0,\widetilde{T}]\) for which \(\dot{x}(t)\) exists and
for all \(t\in [0,\widetilde{T}]\) for which \(\dot{x}(t)\) does not exist.
Then, the convex cone U spanned by
has the property
since otherwise, one could find points on x and C-supporting hyperplanes through these points with the property that the intersection of the C-containing half-spaces bounded by these hyperplanes is nearly bounded (what would imply that x cannot be translated into \(\mathring{C}\)). By the convexity of U, this implies that
and therefore
Since x is a closed characteristic on \(\partial C\), x fulfills \(x(0)=x(\widetilde{T})\). This implies
a contradiction. Therefore, it follows that x cannot be translated into \(\mathring{C}\). \(\square \)
We now consider the operator norm of the complex structure/symplectic matrix J. It is given by:
We derive the following lemma:
Lemma 6.2
Let \(C\subset \mathbb {R}^{2n}\) be a convex body and x a closed characteristic on \(\partial C\) which has period \(\widetilde{T}>0\). Then, we have
Proof
Since x is a closed characteristic on \(\partial C\), we have
This implies
Using Proposition 2.5, we conclude
i.e.,
Therefore, for
we have
and consequently
\(\square \)
Proof of Theorem 1.6
By Theorem 2.7, we have
Let x be an action-minimizing closed characteristic on \(\partial C\), i.e., x fulfills
and minimizes the action with
where we used Euler’s identity (see Proposition 2.3) to derive
Then, since x is in \(\partial C\) and, by Proposition 6.1, cannot be translated into \(\mathring{C}\), (70) together with
(see Proposition 8.2) implies that
Using Lemma 6.2 and (71), we conclude
This implies
Therefore, Viterbo’s conjecture for convex bodies in \(\mathbb {R}^{2n}\) is equivalent to
By referring to Proposition 2.9, we can assume
without loss of generality and get
which, by Theorem 2.7, is equivalent to
By Theorem 3.18, this is equivalent to
and, after applying Proposition 3.10, to
By similar reasoning, Theorem 3.18 also guarantees the equivalence of
and
Moreover, Theorem 3.18 guarantees the following: \(C^*\) is a solution of (73) if and only if \(C^*\) is a solution of (74). \(\square \)
7 Justification of Conjectures 1.8 and 1.9
We start by recalling Conjectures 1.8 and 1.9:
Conjecture
(Conjecture 1.8) We have
Conjecture
(Conjecture 1.9) We have
for \(K\in {\mathcal {C}}(\mathbb {R}^2)\).
We transfer Viterbo’s conjecture onto Wetzel’s problem. For that, we define
and let \(K^*\subset \mathbb {R}^2\) be an arbitrarily chosen convex body of volume y. Then, applying Theorems 2.7 and 3.18, we have
Further, we have
The truth of Viterbo’s conjecture requires
i.e., \(\pi y \geqslant \frac{1}{2}\), which means
Theorem 3.18 also guarantees the sharpness of this estimate.
Together with Theorem 2.7, this justifies the formulation of Conjectures 1.8 and 1.9.
8 Proofs of Theorem 1.11 and Corollary 1.12
We start by recalling Theorem 1.11 and Corollary 1.12:
Theorem
(Theorem 1.11) Let \(K,T\subset \mathbb {R}^n\) be convex bodies. Then, an/the \(\ell _T\)-minimizing closed Minkowski escape path for K has \(\ell _T\)-length \(\alpha ^*\) if and only if \(\alpha ^*\) is the largest \(\alpha \) for which
i.e., for which for every closed path \(\gamma \) of \(\ell _T\)-length \(\leqslant \alpha \), there is a translation \(\mu \) such that K covers \(\mu \left( \{\gamma \}\right) \).
Corollary
(Corollary 1.12) Let \(K,T\subset \mathbb {R}^n\) be convex bodies, where T is additionally assumed to be strictly convex. An/The \(\ell _T\)-minimizing closed (K, T)-Minkowski billiard trajectory has \(\ell _T\)-length \(\alpha ^*\) if and only if \(\alpha ^*\) is the largest \(\alpha \) for which
In order to prove Theorem 1.11, we start with the two following obvious observations:
Proposition 8.1
Let \(K\subset \mathbb {R}^n\) be a convex body. Then we have
Proof
The statement follows directly by recalling that a closed Minkowski escape path is a closed curve whose all translates intersect \(\partial K\) and therefore, equivalently, cannot be translated into \(\mathring{K}\). \(\square \)
Proposition 8.2
Let \(K,T\subset \mathbb {R}^n\) be convex bodies. Then we have
Proof
Since
it suffices to find for every closed curve \(q\in F^{cc}(K)\) a closed polygonal curve \(\widetilde{q}\in F^{cp}(K)\) with
If q cannot be translated into \(\mathring{K}\), then by the remark beyond [35, Lemma 2.1], there are \(n+1\) points on q that cannot be translated into \(\mathring{K}\). By connecting these points, we obtain a closed polygonal curve in \(F^{cp}(K)\) which we call \(\widetilde{q}\). By the subadditivity of the Minkowski functional, it follows (75). \(\square \)
Based on these propositions, we can prove the analogue to Theorem 1.10:
Proof of Theorem 1.11
We first use Proposition 8.1 in order to reduce the statement of Theorem 1.11 to: An/The \(\ell _T\)-minimizing closed curve in \(F^{cc}(K)\) has \(\ell _T\)-length \(\alpha ^*\) if and only if \(\alpha ^*\) is the largest \(\alpha \) for which
First, let us asssume that \(\alpha ^*\) is the \(\ell _T\)-length of an/the \(\ell _T\)-minimizing closed curve in \(F^{cc}(K)\). Then, from Proposition 8.2, we know that there is a closed polygonal curve
i.e., \(q^*\) is a minimizer of
Then it follows from Proposition that
Let \(\alpha > \alpha ^*\). If
then
i.e., every closed curve of \(\ell _T\)-length \(\alpha \) can be covered by a translate of K. This implies that every closed curve of \(\ell _T\)-length \(\lambda \alpha \), \(\lambda <1\), can be covered by a translate of \(\mathring{K}\). From this we conclude
Therefore, there is no \(\alpha > \alpha ^*\) for which (77) is fulfilled, i.e., \(\alpha ^*\) is the largest \(\alpha \) for which (76) holds.
Conversely, if \(\alpha ^*\) is the largest \(\alpha \) for which (76) holds. Then, there is a closed curve \(q^*\) with
Otherwise, if not, then one has
for all closed curves q of \(\ell _T\)-length \(\alpha ^*\). This implies
for all closed curves q of \(\ell _T\)-length \(\alpha ^*\). But then there is a \(\lambda >1\) such that
for all closed curves of \(\ell _T\)-length \(\alpha ^*\). But this is a contradiction to the fact that \(\alpha ^*\) is the largest \(\alpha \) for which (76) holds.
Now, if
and \(\widetilde{q}\) is a minimizer of the left side, then it follows
because, due to Proposition 3.2, with \(\widetilde{\alpha }<\alpha ^*\) one has
Then, with Lemma 3.16, there is a \(\lambda >1\) such that
with
But this is a contradiction to the fact that every closed curve of \(\ell _T\)-length \(\leqslant \alpha ^*\) can be covered by a translate of K. Therefore, it follows
and together with (78), we conclude that
\(\square \)
The proof of Corollary 1.12 follows immediately:
Proof of Corollary 1.12
The proof follows directly by combining Proposition 8.2, [36, Theorem 3.12], and Theorem 1.11. \(\square \)
9 Computational approach for improving the lower bound in Wetzel’s problem
In this section, we aim to present a computational approach for improving the best lower bound in Wetzel’s problem, which, as stated in Theorem 1.7, is due to Wetzel himself (see [52]). But not only that, our approach most likely also allows to find, more generally, lower bounds in Minkowski worm problems. By Theorem 3.18, these lower bounds eventually translate into upper bounds for systolic Minkowski billiard inequalities as well as for Viterbo’s conjecture for convex Lagrangian products.
The main idea of this approach is inspired by a series of works related to the search for area-minimizing convex hulls of closed curves in the plane which are allowed to be translated and rotated. Since the area-minimizing convex cover for a set of closed curves is, equivalently, the area-minimizing convex hull of these closed curves (note that this observation has already used within the proof of Corollary 1.2), these works treat the question of lower bounds for the following version of Moser’s worm problem in which closed arcs are considered:
Find a/the convex set of least area that contains a congruent copy of each closed arc in the plane of length one.
In [11] (applying results from [16]), the first lower bound for the area was found considering the convex hull of a circle and a line segment. In [20], this lower bound was improved by first considering a circle and a certain rectangle and later a circle and a curvilinear rectangle. The latest improvements are due to Grechuk and Som-am who in [24] considered the convex hull of a circle, an equilateral triangle and a certain rectangle, and in [25] the convex hull of a circle, a certain rectangle, and a line segment. However, in order to adapt these approaches to our setting, in the details, we have to make some changes.
But let us first start with some underlying considerations (as in the proof of Corollary 1.2) in the most general case: For arbitrary convex body \(T\subset \mathbb {R}^n\), we ask for lower bounds of
By referring to the above mentioned main idea, we start by noting that for
we have
This estimate can be further improved by
so that, eventually, we get
where the minimum on the left runs for every \(q\in L_T(1)\) over all possible translations in \(\mathbb {R}^n\).
Let us now exemplary show how (80) can be used to calculate lower bounds of (79) within the setting of Wetzel’s problem, i.e., \(n=2\) and \(T=B_1^2\).
Let \(q_1\) be the boundary of \(B_{\frac{1}{2\pi }}^2\),
the boundary of an equilateral triangle \(T_{t_1,t_2,\frac{1}{3},\theta }\) with mass point \((t_1,t_2)\), side length \(\frac{1}{3}\), and angle \(\theta \) between one of the sides and the horizontal line, and let
be the boundary of a rectangle \(R_{r_1,r_2,1,\widehat{q}}\) with middle point \((r_1,r_2)\), perimeter 1, and quotient of the side lengths \(\widehat{q}\).
Then, by definition, we have
for all
and (80) (because of \(\theta \in \left[ 0,\frac{3\pi }{4}\right] \) and \(\widehat{q}>0\), one has \(k=\infty \)) becomes
Then, one can define
which is a convex function with respect to the first four coordinates \((t_1,t_2,r_1,r_2)\) (this can be shown similar to in [24]) and compute
We leave it at that, starting with (80), gives us the ability to tackle many different Minkowski worm problems–in any dimension, for many different Ts and by using diverse closed curves
Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Notes
We round all decimal numbers up to the fifth decimal place.
This implies that q is differentiable almost everywhere with \(\dot{q}\in L^2([0,\widetilde{T}],\mathbb {R}^n)\).
In Proposition 3.9, we will prove that in fact there exists at least one minimizer.
This definition is in fact the outcome of a historically grown study of symplectic capacities. Traced back–recalling that \(c_{EHZ}\) in its present form is the generalization of a symplectic capacity by Künzle in [38] after applying the dual action functional introduced by Clarke in [12], the EHZ-capacity denotes the coincidence of the Ekeland-Hofer- and Hofer-Zehnder-capacities, originally constructed in [15] and [28], respectively.
In this paper, whenever we write products of the form \(K\times T\) for two convex bodies \(K,T\subset \mathbb {R}^n\), we presume the natural symplectic structure of \(\mathbb {R}^{2n}=\mathbb {R}^n_q \times \mathbb {R}^n_p\). So, every such product is a Lagrangian product.
Here, we note that K has been dissolved by replacing it by an expression that extremizes over all possible Ks. The extremizing K is of the form (3).
For the sake of simplicity, whenever we talk of the vertices \(q_1,...,q_m\) of a closed polygonal curve, we assume that they satisfy \(q_j\ne q_{j+1}\) and \(q_j\) is not contained in the line segment connecting \(q_{j-1}\) and \(q_{j+1}\) for all \(j\in \{1,...,m\}\). Furthermore, whenever we settle indices 1, ..., m, then the indices in \(\mathbb {Z}\) will be considered as indices modulo m.
Here, by \(C\times C^\circ \) we denote the Lagrangian product of C and \(C^\circ \), where we presume the natural symplectic structure on \(\mathbb {R}^{2(2n)}\cong \mathbb {R}^{2n}_{q'}\times \mathbb {R}^{2n}_{p'}\) with \(\mathbb {R}^{2n}_{q'}=\mathbb {R}^n_q\times \mathbb {R}^n_p\) and denote by \(q'\) and \(p'\) the local and momentum coordinates on \(\mathbb {R}^{2n}\supset C\), respectively.
Whenever we write
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(K)=c}\;\min _{q\in F^{cp}(K)}\ell _T(q) \end{aligned}$$the maximum is understood to consider only convex bodies \(K\subset \mathbb {R}^n\). This is implicitly indicated by the fact that we defined \(F^{cp}(K)\) only for convex bodies \(K\subset \mathbb {R}^n\).
Whenever we write
$$\begin{aligned} \min _{{{\,\textrm{vol}\,}}(T)=d} \; \min _{K\in A(T,\alpha )} {{\,\textrm{vol}\,}}(K) \end{aligned}$$or
$$\begin{aligned} \max _{{{\,\textrm{vol}\,}}(T)=d}\; \max _{{{\,\textrm{vol}\,}}(K)=c} \; \min _{q\in F^{cp}(K)} \ell _T(q) \end{aligned}$$the minimum/maximum is understood to consider only convex bodies \(T\subset \mathbb {R}^n\). This is implicitly indicated by the fact that we defined \(A(\cdot ,\alpha )\) and \(\ell _{\cdot }(q)\) only for convex bodies \(T\subset \mathbb {R}^n\).
Here, we note that K has been dissolved by replacing it by an expression that extremizes over all possible Ks. The extremizing K is of the form (3).
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Acknowledgements
This research is partly supported by the SFB/TRR 191 ’Symplectic Structures in Geometry, Algebra and Dynamics’, funded by the German Research Foundation, and was carried out under the supervision of Alberto Abbondandolo (Ruhr-Universität Bochum). The author is thankful to the supervisor’s support.
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This research is supported by the SFB/TRR 191 ‘Symplectic Structures in Geometry, Algebra and Dynamics’, funded by the ‘German Research Foundation’ (DFG).
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Rudolf, D. Viterbo’s conjecture as a worm problem. Monatsh Math 201, 217–287 (2023). https://doi.org/10.1007/s00605-022-01806-x
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DOI: https://doi.org/10.1007/s00605-022-01806-x