Bounds on convex bodies in pairwise intersecting Minkowski arrangement of order μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}

The μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-kernel of an o-symmetric convex body is obtained by shrinking the body about its center by a factor of μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}. As a generalization of pairwise intersecting Minkowski arrangement of o-symmetric convex bodies, we can define the pairwise intersecting Minkowski arrangement of order μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}. Here, the homothetic copies of an o-symmetric convex body are so that none of their interiors intersect the μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}-kernel of any other. We give general upper and lower bounds on the cardinality of such arrangements, and study two special cases: For d-dimensional translates in classical pairwise intersecting Minkowski arrangement we prove that the sharp upper bound is 3d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3^d$$\end{document}. The case μ=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =1$$\end{document} is the Bezdek–Pach Conjecture, which asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb R^d$$\end{document} is 2d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^d$$\end{document}. We verify the conjecture on the plane, that is, when d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=2$$\end{document}. Indeed, we show that the number in question is four for any planar convex body.


Introduction
A positive homothetic copy of a convex body (i.e. a compact convex set with non-empty interior) K in Euclidean d-space R d is a set of the form λK + t where λ > 0 and t ∈ R d . Two sets in R d are said to touch each other if they intersect but their interiors are disjoint.
The following notion was introduced by Fejes Tóth [10]: Pairwise intersecting homothets of a centrally symmetric convex body in the d-dimensional Euclidean space form a Minkowski arrangement if none of them contains the Polyanskii [19] recently proved that such a family of convex bodies has at most 3 d+1 members. This result was improved by Naszódi and Swanepoel [16] showing an upper bound of 2·3 d . It is natural to conjecture that the maximum number of elements is 3 d .
We prove the following upper bound on the cardinality of a family containing translates of a centrally symmetric convex body in pairwise intersecting Minkowski arrangement in Sect. 2:

a pairwise intersecting Minkowski arrangement consisting of translates of a centrally symmetric convex body K contains at most 3 d elements. This bound is sharp, equality holds if and only if K is a d-dimensional parallelotope.
We show a construction for arbitrary centrally symmetric convex body that gives a linear lower bound on the cardinality of maximal pairwise intersecting Minkowski arrangements of translates. Although from Theorem 3 of [1] we can deduce the existence of an exponential lower bound, we now give a simple and deterministic construction.

Proposition 1.2. For a centrally symmetric convex body
, a maximum cardinality set consisting of translates of K in pairwise intersecting Minkowski arrangement has at least 2d + 3 elements.
We introduce some generalizations of the problem based on an idea of Fejes Tóth [11], examined by Böröczky and Szabó [6,8]: For 0 ≤ μ ≤ 1 they defined the μ-kernel of an o-symmetric convex body K as μK.
Using this notion, for homothets of an o-symmetric convex body we can consider a pairwise intersecting Minkowski arrangement of order μ, where the homothets are pairwise intersecting but none of their interiors intersect the center of any other.
We prove an upper bound on the cardinality of such an arrangement, then, for centrally symmetric convex bodies we verify the existence of an exponential lower bound. In 1962, Danzer and Grünbaum [9] proved that the maximum cardinality of a family of pairwise touching translates of a convex body K in R d is 2 d , which bound is attained if and only if K is an affine image of a cube. Petty [18] showed that every convex body in the plane (or in 3-space) has three (four) pairwise touching translates. As an extension of this problem, Bezdek and Pach [5] conjectured in 1988 that the maximum number of pairwise touching positive homothetic copies of a convex body in R d is 2 d . They showed that any such family of homothetic copies has at most 3 d elements, and if K is a d-dimensional Euclidean ball, then the maximum is equal to d+2. Naszódi [15] improved the first estimate by proving the upper bound 2 d+1 . In [14], Lángi and Naszódi proved (using a result [4] of Bezdek and Brass about one-sided Hadwiger numbers) the upper bound 3 · 2 d−1 in the case when K is centrally symmetric.
In Sect. 3, we show that the conjecture holds on the plane, moreover, every planar convex body has four pairwise touching homothets.
Theorem 1.5. For any convex body K in R 2 , the maximum number of pairwise touching positive homothetic copies of K is four.
The generalized notion of Minkowski arrangement provides a connection between the original problem of pairwise intersecting Minkowski arrangements and the Bezdek-Pach Conjecture [5]. In both problems we consider pairwise intersecting Minkowski arrangements of order μ, in the first case μ = 0, while in the latter case μ = 1.

Bounds on pairwise intersecting Minkowski arrangements
It is natural to conjecture that in R d , a pairwise intersecting Minkowski arrangement consisting of homothets of a centrally symmetric convex body contains at most 3 d elements. Here we prove this upper bound -and a generalization -for the case when all the homothets in the arrangement are translates of the given body.

Proofs of Theorems 1.1 and 1.3
First, we verify Theorem 1.3, then Theorem 1.1 will follow as a corollary.
Any o-symmetric convex body K can be considered as the unit ball of a normed It is easy to see that having a pairwise intersecting Minkowski arrangement of order μ is equivalent to the following two conditions on the distances between centers: none of them can be farther than 2, nor closer than 1 + μ to any other. After applying a homothety, this is equivalent to the problem when the distances are between 1 and 2 1+μ .
Proof. By the assumption, for different indices the bodies , using the isodiametric inequality for Minkowski spaces [12] we get that Applying this lemma for λ = 2 1+μ , we get that the number of points with pairwise distances between 1 and 2 1+μ is at most 1 + 2 1+μ d , which is equivalent to the statement of Theorem 1.3. To reach this, (2.1) has to hold with two equalities. From the following lemma of Groemer [13] we can see that this happens if and only if K is a d-dimensional parallelotope.

Lemma 2.2.
Suppose that K is a convex body in R d such that for some 1 < t ∈ R the body tK can be decomposed into translates of K. Then K is a d-dimensional parallelotope and t is an integer. The partition is unique. Remark 2.3. The bound in Theorem 1.3 gives the known result 2 d for the pairwise touching case, when μ = 1.

Proof of Proposition 1.2
First, we show a construction of seven bodies in R 2 , then the lower bound 2d + 3 for the higher dimensional cases will follow recursively. In R 2 , consider an affine-regular hexagon inscribed in K that is symmetric about the center of K (see for example [17,Lemma 4.3]). There exist seven translates of this hexagon in Minkowski arrangement, shown in Fig. 1. Translate K in a way that the center points are the same as the centers of the above hexagons. Now a center of any translate is either not contained in another body, or lies on its boundary. Furthermore, these translates share a common point, so they are pairwise intersecting. This means that the construction gives a Minkowski arrangement.
For a centrally symmetric convex body K in R d , denote by M (K) the maximal number of translates in a pairwise intersecting Minkowski arrangement. It is easy to see, that for any K in R 1 , M (K) = 3, and we showed that for K in R 2 , M (K) ≥ 7.
Let e 1 , ..., e d be an Auerbach basis [21,Chapter3] of the space (R d , . K ). In dimension d ≥ 3, using the above planar construction, we can take seven translates of K in a Minkowski arrangement such that their centers lie in the plane of the first two basis vectors e 1 and e 2 : The plane spanned by e 1 and e 2 intersects a translate of K whose center is in this plane in a centrally symmetric convex body K. According to the above construction, we take seven translates of K as in Fig. 1 such that the center of the middle body is o. Now these translates of K are in a planar Minkowski arrangement. Then consider seven translates of K with the same center points as the centers of the seven planar bodies. Since each of their centers lie in the same plane where they are either not contained in another body, or lie on its boundary, these translates of K are also in a Minkowski arrangement. Along each direction e 3 , ..., e d we can add two further translates of K to this arrangement so that they contain o on their boundary (for each i = 3, ..., d, one needs to take the translates of the middle body by ±e i , this only intersects the plane spanned by other two basis vectors on its boundary). Proof of Proposition 1.4. The statement follows from a result of Bourgain [7]. He showed that on the unit sphere of any normed space, there is an exponentially large number of points so that the distance of any two is more than √ 2−ε. Consider the o-symmetric convex body K as the unit ball of the normed space (R d , . K ). Choosing μ < √ 2 − 1, we get exponentially many points on the sphere so that their pairwise distances are between 1+μ and 2. Considering these points as centers, we verify the statement.

Proof of the upper bound in Theorem 1.5
Let K = {K 1 , K 2 , . . . , K n } be a family of pairwise touching positive homothetic copies of a planar convex body K.
If there is a point that belongs to four of the homothets, then we can enlarge (or shrink) each of the four bodies from that point as a center, to obtain four touching translates of K. By the result of Danzer and Grünbaum [9], this implies that K is a parallelogram. It is easy to see that in this case, the family does not have a fifth member. Thus, from this point on, we will assume that no point belongs to four of the homothets.
If there is a point that belongs to three of the homothets and K has at least four members, then we will show that this point also belongs to a fourth body. Proposition 3.1. Let K 1 , K 2 , K 3 , K 4 be pairwise touching positive homothets of the convex body K in R 2 . If Proof. We can assume that K contains the origin in its interior, and that K i = λ i K + x i for some λ i ∈ R and x i ∈ R 2 . Let p ∈ K 1 ∩ K 2 ∩ K 3 , and C i be the smallest angular region with vertex p containing K i .
Suppose that for a pair i = j there exists c ∈ (int C i ∩ int C j ). Then the line pc intersects the interior of both K i and K j because C i and C j are the smallest angular regions containing K i and K j respectively. Hence due to the convexity of the bodies, K i overlaps K j , which is a contradiction.
Suppose that K 1 ∩ K 2 ∩ K 3 ∩ K 4 = ∅. Then p / ∈ K 4 , thus there exists a supporting line of K 4 that does not go through p and separates K 4 from p. K 4 touches K 1 , K 2 and K 3 , hence each of these three bodies has a point in both of the closed half-planes bounded by . From this it follows that intersects the angular regions C 1 , C 2 and C 3 . For every i ∈ {1, 2, 3}, ∩ C i is a connected Vol. 111 (2020) Bounds on convex bodies in pairwise intersecting Page 7 of 11 27 Figure 2 The smallest angular regions containing the bodies subset of , thus there is a middle one of them. Without loss of generality we can assume that this one is K 1 . Let v 1 = p − x 1 . The image of p by the homothety that maps K 1 to K 4 is the point x 4 + λ4 λ1 · v 1 ∈ . The same homothety maps C 1 to the angular region C 1 := C 1 + x 4 − x 1 + v 1 · λ4 λ1 − 1 , see (Fig. 2). As K 1 ⊂ C 1 and the bodies are positive homothets, K 4 ⊂ C 1 follows. At least one pair of the bounding lines of C 1 and C 1 are different, thus due to the fact that int C i ∩ int C j = ∅ for any i = j, i, j ∈ {1, 2, 3} C 1 is disjoint from at least one of the angular regions C 2 and C 3 . But in this case K 4 cannot touch the body lying in this angular region, which is a contradiction.
Thus, it is enough to consider the case when no point belongs to three of the homothets. Proof. For each i ∈ {1, ..., n}, choose an interior point p i ∈ K i . The bodies are pairwise touching, so we can draw a curve between any two of the chosen points p i , p j so that it lies in K i ∪ K j . Since no three of the bodies share a common point, these curves intersect only in the interior of the bodies. It is easy to see that we can eliminate these intersections with a perturbation. This way we draw the complete graph of n vertices on the plane, from which n ≤ 4 follows immediately.

Proof of the lower bound in Theorem 1.5
In this section, we show that for any planar convex body K, there are four pairwise touching homothets of K.
Consider two distinct parallel support lines of K that each touch K at one point: x 1 and x 2 . The existence of such a pair of lines follows from Theorem 2.2.9. of [20], but may also be proved as an exercise.
Let K 1 = K and K 2 = K + x 2 − x 1 . Let f be the line through the single point of contact, x 2 , of K 1 and K 2 parallel to x 2 − x 1 . On both sides of f , there is a translate of K that touches both K 1 and K 2 . Indeed, if we push K around K 1 so that it always touches K 1 then, by continuity, such two positions will be found.
If on both sides we can find such translates of K that also contain x 2 then x 2 is a common point of four translates of K and we are done. Thus we assume that at least one of these translates does not contain x 2 . We call this translate K 3 . Now, K 1 , K 2 , K 3 are pairwise touching translates of K that do not share a common point. It follows that they surround a bounded region R with nonempty interior. Consider the largest homothet K 4 of K contained in R. To finish the proof, we claim that K 4 touches K 1 , K 2 and K 3 . Indeed, assume K 4 touches only two of them, say K 1 and K 2 . Consider a line that separates K 4 and K 1 , and another line that separates K 4 and K 2 . Let u 1 and u 2 be the unit normal vectors of these two lines respectively, pointing towards K 4 . Clearly, if u 1 is not parallel to u 2 , then K 4 can be moved a little inside R (in the direction of u 1 + u 2 ) so that it does not touch either K 1 , K 2 or K 3 . Then, we may enlarge K 4 slightly within R contradicting the maximality of K 4 . Thus u 1 is parallel to u 2 . However, in this case, K 1 and K 2 are strictly separated, which is a contradiction, finishing the proof of the lower bound in Theorem 1.5.

A topological note
In this section we present Proposition 5.1, a topological observation, which may be used in place of Proposition 3.2 to prove the upper bound in Theorem 1.5.
An arc in the plane is the image of an injective continuous map of the [0, 1] interval into the plane. A Jordan curve in the plane is the image of an injective continuous map of the circle into the plane. We will call the closed bounded region bounded by a Jordan curve a Jordan region.