Variational worn stones

We introduce an evolution model \`a la Firey for a convex stone which tumbles on a beach and undertakes an erosion process depending on some variational energy, such as torsional rigidity, principal Dirichlet Laplacian eigenvalue, or Newtonian capacity. Relying on the assumption of existence of a solution to the corresponding parabolic flow, we prove that the stone tends to become asymptotically spherical. Indeed, we identify an ultimate shape of these flows with a smooth convex body whose ground state satisfies an additional boundary condition, and we prove symmetry results for the corresponding overdetermined elliptic problems. Moreover, we extend the analysis to arbitrary convex bodies: we introduce new notions of cone variational measures and we prove that, if such a measure is absolutely continuous with constant density, the underlying body is a ball.


Introduction
Aim of this paper is to propose a variational counterpart of Firey's seminal result stating that the fate of worn stones is that of becoming spherical [25].Firey considered the evolution problem satisfied by the support functions h(t, •) of a family of convex bodies C(t) in R 3 , obtained when an initial convex stone C(0) tumbles on an abrasive plane and undertakes an erosion process, in which the rate of wear is proportional to the density of contact points with the plane per unit surface area, and also to the volume of the stone.The mathematical formulation reads (1) (t, ξ) = −a|C(t)|G(t, ν −1 t (ξ)) on (0, T ) × S n−1 , h(0, ξ) = h 0 (ξ) on S n−1 , h(t, ξ) ≤ b on (0, T ) × S n−1 , where a and b are positive constants, ν −1 t is the inverse Gauss map of C(t), G(t, •) denotes the Gaussian curvature of C(t), and |C(t)| its volume.Under the assumption that C(0) is smooth and centrally symmetric, and that, for some T > 0, the evolution problem (1) admits a smooth solution, Firey showed that actually T = +∞, that convexity and symmetry are preserved along the flow, and that an ultimate shape must satisfy the following geometric condition: in every direction its support function h and Gaussian curvature G are proportional to each other.Precisely, denoting by ν its Gauss map and by c a positive constant, it satisfies (2) h(ξ) = c G(ν −1 (ξ)) ∀ξ ∈ S 2 .
Then Firey obtained the following symmetry result: if a smooth and centrally symmetric convex body in R 3 satisfies the equality (2), it must be a ball.Moreover he conjectured that, in dimension 3, the above results (often referred to as "convergence of solutions to a round point") should remain valid without the assumption of central symmetry; in higher dimensions, this is also called the generalized Firey conjecture.Firey's vision of worn stones has inspired many deep developments up today, in both the fields of evolution equations, and of convex geometry.
On the side of evolution equations, the parabolic problem introduced by Firey, written in terms of a parametrization of the boundary and up to a renormalization, is nothing else than the Gaussian curvature flow.Such flow has been intensively studied in recent years, so that it is impossible to report the related literature.We refer to the recent monograph [4] for an account about the Gaussian curvature flow and more references on it.Let us just mention that nowadays a full resolution is available for both Firey's conjecture and its generalized version, the main steps of its history being the following: the existence of a solution to the Gauss curvature flow was proved in any space dimension by Chou [52]; the convergence to a round point was obtained in dimension n = 3 by Andrews [3], while in higher dimensions it follows from a convergence result by Guan-Ni [29], combined with a rigidity result by Brendle-Choi-Daskalopoulos [17].
On the side of convex geometry, the rigidity result by Firey is related to more general rigidity questions for convex bodies.In fact, condition (2) amounts to ask that a convex body K has the same cone volume measure V K as a ball (see Section 2 for the definition).Accordingly, a generalized version of Firey rigidity problem reads as follows: for K belonging to the class K n of convex bodies in R n , and a ball B, investigate the validity of the implication (3) In the above mentioned papers about the asymptotic behaviour of the Gaussian curvature flow, it has been proved that (3) holds under the assumption that K has its centroid at the origin [29,Proposition 3.3] or that K is smooth and strictly convex [17,Theorem 6].
Looking at (3), one is led in a natural way to the more general question whether the cone volume measure determines uniquely the associated convex body.In particular, denoting by K n * the class of centrally symmetric convex bodies in R n , the question is whether, given K and K 0 in K n * , it holds that (4) The relevance of this question lies in particular in its connection with one of the major open problems in convex geometry, namely the validity of the log-Brunn-Minkowski inequality: ( and h K , h L denote the support function of K and L respectively.Inequality (5), which is a strengthening of the classical Brunn-Minkowsi inequality, has been proved in dimension n = 2 in [13], while it is currently open in higher dimensions (see the recent paper [10] for an extensive survey about the state of the art in this topic, and also our previous paper [22] for a related functional version).As shown in [13], proving the log-Brunn-Minkowski inequality is equivalent to proving that, for every fixed K ∈ K n * , the minimization problem is solved by K itself.Moreover, it turns out that a solution K 0 to problem (6) exists and is a critical set, in the sense that it satisfies the equality of measures V K 0 = V K .Hence the connection with the implication (4).For n = 2, Böröczky-Lutwak-Yang-Zhang [13] proved that such implication holds true, thus obtaining also the log-Brunn-Minkowski inequality.
To close the circle, the minimization problem ( 6) is not unrelated with the evolution of convex bodies by their Gaussian curvature.Indeed, in the particular case K = B, the integral in ( 6) is an entropy functional, firstly considered by Firey, which decreases along the Gaussian curvature flow (and actually a suitable generalization of this entropy monotonicity property is used by Guan-Ni to obtain (3) for bodies with centroid at the origin).
In this work we attack the new problem of studying the above topics when the volume functional is replaced by some variational energy, such as torsional rigidity, principal Dirichlet Laplacian eigenvalue, or Newtonian capacity.These functionals are object of study in many problems of classical and modern Calculus of Variations, such as isoperimetric type inequalities, concavity inequalities, inequalities involving polarity.In many cases, the behaviour of these functionals turn out to resemble that of the volume functional, in particular concerning the validity of inequalities of Brunn-Minkowski type (see [20]).Thus, studying variational results à la Firey seems to be a very natural direction, which is, to the best of our knowledge, completely unexplored.To precise what we intend, let us focus our attention in particular on the case of torsional rigidity.In the physical case when K ⊂ R 2 is the cross section of a cylindrical rod K × R under torsion, its torsional rigidity T (K) is the torque required for unit angle of twist per unit length.From an analytical point of view, when K is more in general a n-dimensional convex body, its torsional rigidity (often abbreviated as torsion) is given by where u K is the torsion function of K, i.e. the unique solution to the Dirichlet problem A classical result by Dahlberg [23] ensures that, for any K ∈ K n , the gradient of the torsion function is well-defined H n−1 -a.e. on ∂K, and belongs to L 2 (∂K).Moreover, it is also well-known that the Hadamard first variation of torsion can be expressed through an integral formula, which is the perfect analogue of the one valid for volume, replacing the surface area measure S K by the torsion first variation measure µ K , defined by ( 7) (ν K )♯ denotes the push forward through the Gauss map of K.By pursuing the analogy with the case of volume, we are led to introduce the cone torsion measure of K, as the positive measure on S n−1 defined by where h K denotes the support function of K.
Relying on this new definition, it is natural to investigate the rigidity question analogue to the one of Firey, namely whether, for a given In the setting of smooth convex bodies, the equality τ K = τ B amounts to ask that the torsion function u K satisfies, for some positive constant c, the boundary condition where G K denotes the Gaussian curvature of ∂K.Hence, the corresponding rigidity problem consists in investigating symmetry of smooth convex bodies K such that the following overdetermined boundary value problem admits a solution: This problem is not covered by the very vast literature on Serrin-type problems [53], as the simultaneous presence of the support function and of curvature in the overdetermined boundary condition is completely new; we limit ourselves to quote the papers [15,26] about overdetermined problems in which curvatures are involved.In Theorem 3 we establish symmetry for problem (8), under the weaker assumption that K has its centroid at the origin.At present, we do not know if the assumption on the centroid can be removed.The proof of Theorem 3 is obtained by an ad-hoc combination of different inequalities, such as the Saint-Venant inequality for torsion, and the isoperimetric inequality for the 2-affine surface area (or alternatively, an isoperimetric-type inequality proved in [19] and Blasckhe-Santaló inequality).In the setting of arbitrary convex bodies, the equality τ K = τ B cannot be formulated any longer as a pointwise equality for the gradient of the torsion function along the boundary; it just tells the measure τ K is a constant multiple of the diffuse measure on the sphere, i.e. (9) τ K = c H n−1 S n−1 .
In Theorem 6 we establish symmetry for convex bodies with centroid at the origin satisfying (9).We argue by adapting the arguments used in the regular case, but the proof is more delicate, as it involves the regularity properties of convex bodies having an absolutely continuous surface area measure; in this respect, we heavily rely on the results proved by Hug in the papers [32][33][34].
The symmetry results described so far are related to many further questions.In particular, in the spirit of Firey's work, it is natural to wonder whether the condition τ K = τ B identifies K as the ultimate shape of some flow.
To answer this question, we imagine a new variational flow for worn stones which tumble on an abrasive plane and undertake an energy-based erosion process.Namely, we assume that the rate of wear is proportional to the density of contact points with the abrasive plane, no longer per unit surface area measure as in [25], but per unit torsion first variation measure; moreover, while in [25] the rate of wear is also taken to be proportional to the volume, we take it to be proportional to the torsional rigidity.
To write explicitly the problem, let C(t) represent the evolution in time of an initial convex stone C(0), and let σ be any small part of ∂C(t).The measure of the set of directions for which the abrasive plane touches C(t) at points in σ is given by H n−1 (ν t (σ)), where ν t is the Gauss map of C(t).By a standard change of variables (see Proposition 7 (e)), we have where G(t, y) denotes the Gauss curvature of ∂C(t) at the point y.Recalling the definitions of surface area measure and of torsion first variation measure for the convex body C(t), the above integral can also be written as , where u(t, •) is the torsion function of C(t).Thus we see that, while contact points have density G(t, ν −1 t (ξ)) with respect to the surface area measure S C(t) , they have density 2 with respect to the first variation measure µ C(t) in (7).We conclude that the variational analogue of Firey's problem when replacing volume by torsion reads as follows: (10) As in Firey's paper, we assume that C(0) is smooth and centrally symmetric, and that for some T > 0, the evolution problem (10) admits a smooth solution.Under this assumption, Theorem 10 establishes that T = +∞, that convexity and symmetry are preserved along the flow, and that an ultimate shape has the same cone torsion measure as a ball.The proof relies basically on the Brunn-Minkowski inequality for torsion due to Borell [8].
We remark that proving the existence of a unique smooth solution to problem ( 10) is an interesting problem in parabolic flows, which is beyond the scopes of this paper, and will be object of further research.
To conclude, another question we wish to address is whether, by analogy with (4), for any pair of convex bodies K, K 0 ∈ K n * , we have (11) τ Such implication is in turn related to the possibility of strengthening the Brunn-Minkowski inequality for torsion into a logarithmic Brunn-Minkowski inequality of the kind . Indeed, by arguing as in the proof of Lemma 3.2 in [13], one can easily check that ( 12) is equivalent to proving that, for every fixed K ∈ K n * , the minimization problem is solved by K itself; moreover, if the above problem admits solution K 0 , it must satisfy the first order optimality condition τ K 0 = τ K (cf. the proof of Theorem 7.1 in [13]).The inequality (12) (or the implication ( 11)) is likely very challenging, and it should be first proved (or disproved) in dimension n = 2, see Section 6 for some related remarks.
To conclude, we point out that all the results discussed so far are valid also when torsional rigidity is replaced either by the principal frequency of the Dirichlet Laplacian, or (for n ≥ 3) by the Newtonian capacity.To some extent, this may lead to consider the analysis our variational worn stones more righteous, and to believe that the corresponding parabolic flows and logarithmic inequalities are worth of further investigation.
Outline of the paper.In Section 2 we detail the definitions of cone variational measures and related representation formulas.In Section 3 we prove symmetry for smooth centred convex bodies K where overdetermined boundary value problems such as (8) admit a solution.In Section 4 we extend the rigidity results of the previous section to the general framework of arbitrary centred convex bodies.In Section 5 we relate the convex bodies studied in the previous sections to new variational flows.In Section 6 we give a few concluding remarks about logarithmic Brunn-Minkowski-type inequalities.

Cone variational measures and related representation formulas
Let K n be the class of convex bodies in R n with the origin in their interior.In particular, we are going to consider convex bodies in K n whose centroid coincides with the origin.Here and below, we indicate by | • | the volume functional on K n .We shall also consider the subclass K n * of centrally symmetric convex bodies.Given K ∈ K n , we denote by h K and ν K respectively its support function and Gauss map.By definition, if m is a measure on ∂K, its push-forward (ν K ) ♯ (m) through the Gauss map is the measure on S n−1 defined by We recall that the surface area measure S K and the cone volume measure V K are defined respectively by moreover, denoting by |V K | the total variation of V K , it holds that The name cone volume measure is motivated by the fact that, when K is a polytope with facets F i and unit outer normals ν i , it holds that ( 14) where δ ν i is a Dirac mass at ν i and ∆(o, F i ) is the cone with apex o and basis F i .In recent years, this notion of cone volume measure has been widely studied (see [5, 11-14, 28, 38, 39, 42, 44-46, 51, 55, 56]).Let now F (K) be one of the following variational energies: torsional rigidity T (K), first Dirichlet Laplacian eigenvalue λ 1 (K), or Newtonian capacity cap(K) (the latter in dimension n ≥ 3).We denote by u K the corresponding ground state, defined as the unique solution in Ω = intK to the following elliptic boundary value problems (in the second case normalized so to have unit L 2 -norm): It is well known that, by analogy with the case of volume, each of these functionals and its Hadamard derivative satisfy the following representation formulas (see respectively [21], [35] and [36]) Here α denotes the homogeneity degree of F under domain dilation (which equals equal to n + 2 for torsion, −2 for the eigenvalue, and n − 2 for capacity) and µ K is the first variation measure of F , defined by ( 15) We set the following: Definition 1 (cone variational measures).For any of the functionals above, with any K ∈ K n we associate the positive measure on S n−1 defined by h K µ K , being µ K given by ( 15), and we denote it respectively by τ K (cone torsion measure), by σ K (cone eigenvalue measure) and by η K (cone capacitary measure).
Let us point out that the terminology in Definition 1 is chosen to emphasize the analogy with the case of volume, though this is somehow an abuse; indeed, the analogue of the equality ( 14) is clearly false for our variational funtionals, since they are not additive under the decomposition of a polytope in K n into cones with apex o and bases at its facets.
Let us also notice that the above representation formulas can be rewitten as integrals over the boundary and in terms of the total variations of the corresponding cone variational measure, respectively as ( 16) Remark 2. The cone variational measures introduced in Definition 1 are weakly * continuous with respect to the convergence of convex bodies in Hausdorff distance.Indeed, if K n converge to K ∞ in Hausdorff distance, the support functions of K n converge uniformly to the support function of K ∞ , while the first variation measures of K n converge weakly * to the first variation measure of K ∞ (cf.respectively [36,Thm. 3.1] for capacity, [35,Section 7] for the first eigenvalue, and [21, Thm.6] for torsion).

Rigidity results for smooth convex bodies
In this section we deal with overdetermined boundary value problems on smooth convex bodies K, for which the Gauss map ν K and the Gaussian curvature G K can be classically defined.Whenever no confusion may arise, we omit the index K and we simply write ν and G.We denote by B a generic ball, by B 1 the unit ball, and by ω n its Lebesgue measure.
Theorem 3. Let K ∈ K n have its centroid at the origin, and boundary of class C 2 .Assume that, for some constant c > 0, there exists a solution to the following overdetermined boundary value problem on Ω := intK: Then K is a ball.
The proof of Theorem 3 is based on the application of different inequalities for convex bodies, some involving torsional rigidity and some others being purely geometrical.We list them below: (i) The Saint-Venant inequality for torsion [31,47]: , with equality if and only if K is a ball.More explicitly, since T (B 1 ) = ωn n(n+2) , ( 19) with equality if and only if K is a ball.
(ii) The isoperimetric inequality of the p-affine surface area in the case p = 2: with equality if and only if K is an ellipsoid (see [41,  (iii) The isoperimetric-type inequality: (iv) The Blaschke-Santaló inequality [6,48] for convex bodies with centroid at the origin: Proof of Theorem 3. We are going to obtain an upper bound and a lower bound for the constant c appearing in the overdetermined boundary condition, starting from two different expressions of it.For the upper bound we use inequality (i), while the lower bound can be obtained by two different methods: either by using the inequality (ii), or by using the inequalities (iii)-(iv).We provide both methods because, as we shall see, the first one works to extend the result to Newtonian capacity, while the second one works for the first Laplacian Dirichlet eigenvalue.Since the upper bound and the lower bound turn out to match each other, we conclude that in particular Saint-Venant inequality (19) holds as an equality, and hence K must be a ball.
-Upper bound for c.We integrate over the boundary both sides of the pointwise overdetermined condition By using respectively the identity ( 16) and the change of variables formula (2.5.29) in [49], we obtain Hence we get Then by using the Saint-Venant inequality (19) we obtain the upper bound , with equality if and only if K is a ball.
-Lower bound for c (first method).We integrate on ∂K the pointwise overdetermined condition, after rewriting it as on ∂K .
Exploiting the fact that u solves the torsion problem, we obtain We now look at the integral appearing the denominator in the right-hand side of (26).Let us distinguish the cases n = 2 and n ≥ 3.
If n = 2, such integral is exactly the 2-affine surface area: If n ≥ 3, the same integral can be estimated in terms of the 2-affine surface area using Hölder's inequality.Specifically, using Hölder's inequality with the conjugate exponents Hence, for every n ≥ 2, we have that where in the second inequality we have used the 2-affine isoperimetric inequality (20).From ( 26) and ( 27), we get .
-Lower bound for c (second method).We integrate over ∂K the pointwise overdetermined condition, after rewriting it as Then, using also the expression of T (K) in (24), the isoperimetric inequality (21) reads: Rising the above inequality to power (n + 2), and setting for brevity We now look at the integral appearing the denominator in the right-hand side of (29).We transform it into an integral on S n−1 by the classical change of variables formula already quoted above.Then, by using Hölder's inequality with conjugate exponents n and n n−1 , the well-known representation formula for the volume of the dual body |K o | = 1 n S n−1 h −n K , and the Blaschke-Santaló inequality (22), we obtain . We now combine ( 29) and (30).Taking also into account that T (B 1 ) = ωn n(n+2) , and we arrive exactly at the inequality (28).
-Conclusion.By comparing the upper bound (25) and the lower bound (28), we conclude that the equality sign holds in both inequalities.In particular, the equality sign in (25) implies that K is a ball.
Theorem 4. Let K ∈ K n have its centroid at the origin, and boundary of class C 2 .Assume that, for some constant c > 0, there exists a solution to the following overdetermined boundary value problem on Then K is a ball.
The proof of Theorem 4 is similar to the one of Theorem 3; it is based on the following inequalities: (i) The Faber-Krahn inequality (ii) The isoperimetric-type inequality (iii) The Blaschke-Santaló inequality (22).
Proof of Theorem 4. Similarly as in the proof of Theorem 3, we are going to provide a matching upper and lower bound for the constant c appearing in the overdetermined boundary condition.A lower bound on c is obtained by arguing as done to obtain an upper bound in the proof of Theorem 3. First we integrate over ∂K the overdetermined condition written as Using the identity (17), and a change of variables as in (23), we deduce that Then, by using the Faber-Krahn inequality (31), we obtain the lower bound with equality if and only if K is a ball.An upper bound on c is obtained by arguing as done to obtain a lower bound in the proof of Theorem 3, second method.Namely, we integrate on ∂K the overdetermined condition, after rewriting it as By using also the expression of λ 1 (K) in (33), the isoperimetric inequality (32) reads: ∂B |∇u B | 2 .Rising the above inequality to power 2, and setting for brevity For the integral appearing at the right hand side of the above inequality, the estimate (30) found in the proof of Theorem 3 holds.Hence we get (35) c ≤ 8 n ω To conclude the proof, it remains to show that the expressions at the right hand sides of the lower bound (34) and of the the upper bound (35) coincide.Indeed in this case the Faber-Krahn inequality must hold as an equality, yielding that K is a ball.The matching of our upper and lower bounds corresponds to the equality The validity of ( 36) is checked through some direct computations involving Bessel functions, that we enclose for the sake of completeness.We have (see for instance [37, Section 4]) (37) λ 1 (B 1 ) = (j n 2 −1,1 ) 2 , where j n 2 −1,1 is the first zero of the Bessel function J n 2 −1 , and ) .The value of the constant C is determined by imposing that u B 1 has unit L 2 -norm, yielding (here and in the sequel we have written for brevity J := J n 2 −1 and j 1 := j n 2 −1,1 ).Since, by known properties of Bessel functions (see [1, 11.4.5])we have we infer that From ( 37) and ( 38), we see that ( 36) is satisfied and our proof is achived.
Theorem 5. Let K ∈ K n have its centroid at the origin, and boundary of class C 2 .Assume that, for some constant c > 0, there exists a solution to the following overdetermined boundary value problem on the complement of K: Then K is a ball.
Also the proof of Theorem 5 follows the same strategy of Theorem 3. It is based on the following inequalities: (i) The isoperimetric inequality for capacity [47] cap(K) , with equality if and only if K is a ball.More explicitly, since cap with equality if and only if K is a ball.(ii) The isoperimetric inequality (20) for the p-affine surface area in the case p = 2.
Proof of Theorem 5. A lower bound for c is obtained by arguing as done to obtain an upper bound in the proof of Theorem 3. Specifically, by ( 18) and ( 23), we obtain Then, by using the isoperimetric inequality (39) for capacity, we get with equality if and only if K is a ball.
An upper bound for c is obtained by arguing as done to obtain a lower bound in the proof of Theorem 3, first method.We integrate over ∂K the overdetermined condition, after rewriting it as .
Using the equalities cap(K) = ∂K |∇u| (see [27, p.27]) and ( 40), we obtain For the integral appearing at the right hand side of the above inequality, the estimate (27) found in the proof of Theorem 3 (through the use of Hölder inequality and the 2-affine isoperimetric inequality) holds.Thus we get Comparing ( 41) and ( 43), we see that our lower and upper bounds match each other, implying in particular that (41) must hold with equality sign, and hence that K is a ball.

Rigidity results for arbitrary convex bodies
In this section we drop any smoothness assumption and we deal with arbitrary centred convex bodies having some cone variational measure equal to that of a ball.Theorem 6.Let K ∈ K n have its centroid at the origin, and let τ K be its cone torsion measure according to Definition 1. Assume that, for some positive constant c, Then K is a ball.
The proof of Theorem 6 requires some preliminaries about convex bodies which have absolutely continuous surface area measure (or equivalently which admit a curvature function); we gather them in Proposition 7, relying on some results proved in [32][33][34].We recall that, if a convex body K has absolutely continuous surface area measure, by definition there exists a unique non-negative function , called curvature function of K, such that its surface area measure satisfies Equivalently, we have Below we follow the usual convention that ν : ∂K → S n−1 is a possibly multivalued map associating to every x ∈ ∂K the unit vectors of the normal cone to ∂K at x.Moreover, using the notations in the cited papers of Hug, we set -M(K):= the set of points x ∈ ∂K such that ∂K is second order differentiable at x (so that ν(x) is a singleton and G(x) is well-defined as the product of the principal curvatures); -(∂K) + := the set of points x ∈ ∂K such that there exists an internal tangent ball touching ∂K at x (so that G(x) is finite); -exp * K:= the set of points x ∈ ∂K such that there exists a closed ball B containing K with x ∈ ∂B (so that G(x) > 0).
By (45) and Theorem 2.3 in [34], it follows that C K is absolutely continuous with respect to To prove statement (d), we consider the subset of S n−1 given by Σ := ξ ∈ ν(R(K)) : h K is second order differentiable at ξ , and we replace R(K) with the possibly smaller set ν −1 (Σ).Then statement (d) follows from (c) and Alexandroff theorem.Finally, let us prove (e).Following [32], for every r > 0 let (∂K) r be the set of all points of x ∈ ∂K such that there exists an internal tangent ball of radius r touching ∂K at x.
We are now in a position to give the Proof of Theorem 6.The idea is to follow the same proof line of Theorem 3.However, this cannot be done directly, since we do not have any longer a pointwise identity holding along the boundary.So, we have to prove first of all that the constant c appearing in ( 44) can be still expressed by the two different formulas ( 24) and (26).To obtain formula (24), it is enough to observe that the two measures in the overdetermined conditions must have the same total variation: using ( 16), we obtain and hence To obtain also formula (26), we need first to prove the following Claim: the equality (44) implies that K has absolutely continuous surface area measure, and in addition its curvature function is strictly positive.This amount to show that, if the equality (44) holds, for a Borel set E ⊂ S n−1 , the following implications hold: (46) Indeed, (46) implies that K admits a curvature function f K ; then, since Recalling that, by definition, we have and ) , the implication (46) follows from the fact that Indeed, the term x•ν remains strictly positive since by assumption K contains the origin in its interior, while the term |∇u| 2 remains strictly positive by Hopf boundary point lemma (since K admits an inner touching ball at H n−1 -a.e.x ∈ ∂K, see e.g.[43]).Let E ⊂ S n−1 be a Borel set with H n−1 S n−1 (E) > 0. From (44), it follows that τ K (E) = c H n−1 (E) > 0. In turn, by (48), this implies that H n−1 (ν −1 (E)) > 0, or equivalently that S K (E) > 0, proving (47).
Since we have just proved that K admits a positive curvature function f K , recalling also the definition of cone torsion measure τ K , the equality (44) can be reformulated as Namely, for every Borel set E ⊂ S n−1 , it holds that (49) We now invoke Proposition 7. By statements (a) and (d), we can rewrite the right hand side of the above equality as so that (49) turns into (50) By Proposition 7, statements (b) and (c), we know that ν R(K) is a bijection from R(K) to ν(R(K)), with H n−1 (∂K \ R(K)) = 0 and H n−1 (S n−1 \ ν(R(K))) = 0.In view of this fact, the integral equality (50) yields the pointwise equality H n−1 -a.e. on ∂K .
Integrating over ∂K, we see that (26) holds.
Having established the validity of ( 24) and ( 26), the remaining of the proof can proceed in the analogous way as in Theorem 3. Indeed, the inequalities used in the proof of Theorem 3 do not need any smoothness assumption.In particular, for the 2-affine isoperimetric inequality for convex bodies which admit a curvature function, we refer to [38,Theorem 3] (see also [32,Remark p. 296]).
As well as the inequalities used in the proof of Theorem 3, also the inequalities used in the proof of Theorems 4 and 5 do not need any smoothness assumption.Thus we obtain the following results for convex bodies with the same cone eigenvalue measure or the same cone capacity measure as a ball.We omit their proofs since they are analogous to the one of Theorem 6 detailed above.
Theorem 8. Let K ∈ K n have its centroid at the origin, and let σ K be its cone eigenvalue measure according to Definition 1. Assume that, for some positive constant c, Then K is a ball.
Theorem 9. Let K ∈ K n , n ≥ 3, have its centroid at the origin, and let η K be its cone capacity measure according to Definition 1. Assume that, for some positive constant c, Then K is a ball.

Ultimate shapes of variational flows
In this section we show that, in the smooth centrally symmetric setting, the convex bodies analysed in Section 3 can be identified with the ultimate shapes of new variational flows, under the basic assumption that they admit a solution.We present a unified treatment, by considering the evolution problem ( 51) where F may denote either the torsional rigidity, or the principal Dirichlet Laplacian eigenvalue, or the Newtonian capacity (the latter for n ≥ 3), and u is the corresponding ground state.Let us remark that, in terms of the parametrization of ∂C(t) by its inverse Gauss map , the evolution equation in ( 51) can be written as (indeed, we have h(t, ξ) = ν −1 t (ξ) • ξ, so that ∂h ∂t (t, ξ) = ∂X ∂t (t, ν −1 t (ξ)) • ξ).Thus, up to the factor F (C(t)) (which is just a rescaling allowing to have global existence in time), the unique crucial difference bewteen (51) and the classical Gaussian curvature flow is the presence of the squared modulus of the ground state gradient.We denote by α the homogeneity degree of F under domain dilations (see Section 4).The following statement deals with the cases of torsion and first eigenvalue, see however Remark 11 for the case of capacity.
Theorem 10.Let F = T or F = λ 1 .Assume that, for some T > 0, there exists a unique family of convex bodies C(t), defined for t ∈ [0, T ), which are smooth (at least of class C 3 ) and such that their support function ξ → h(t, ξ) satisfies (51).Then, setting γ = anωn |α| , (i) it holds that F (t) = F 0 exp(−γt), where F 0 is the energy at t = 0; in particular, the family C(t) is strictly decreasing by inclusion; (ii) the flow is defined for every t ≥ 0 (i.e., T = +∞), and the sets C(t) are strictly convex bodies; (iii) the following entropy is decreasing along the flow: (iv) assuming in addition that C(0) is centrally symmetric, we have that C(t) is centrally symmetric for every t ≥ 0; moreover, if {t n } → +∞, up to subsequences exp(γt n )C(t n ) converge in Hausdorff distance to a centrally symmetric convex body, called an ultimate shape for problem (10), whose cone torsion measure or cone eigenvalue measure is absolutely continuous with respect to H n−1 S n−1 , with constant density.
Remark 11.As it can be seen by direct inspection of the proof below, for F = cap, thanks to the Brunn-Minkowski inequality proved in [7], statement (i) continues to hold, but in principle statement (ii) may fail, because the positivity of the capacity does not imply that the corresponding convex body is nondegenerate.However, if it occurs that the flow is defined for all times, then also statements (iii)-(iv) hold true.
(i) Set where µ K is the first variation measure given by ( 15); recall that, by the Brunn-Minkowski inequality satisfied by F (see [9] for torsion and [16] for the first eigenvalue), it holds that (52) Rewrite the equation as Integrating over S n−1 with respect to ξ, we obtain where we used for shortness the notation F 1 (t, t ′ ) := F 1 (C(t), C(t ′ )), F (t) := F (C(t)).Then, using (52), we arrive at Consider now the convex body C(t) with support function h(t, ξ) := h(t, ξ) exp(γt).Denoting by u, ν t , G, F respectively its torsion function, Gauss map, Gaussian curvature, and torsional rigidity, it holds that Taking γ = anωn |α| , by (53), we have The equation satisfied by h is Integrating this equation on S n−1 , by the choice of γ and the representation formula for F (t), we infer that Consequently, differentiating under the sign of integral (thanks to the smoothness assumption made on C(t) and standard elliptic boundary regularity), that we may rewrite as where we have set F 1 (t + s, t) := F 1 ( C(t + s), C(t)).Then, using (52) in a similar way as above, we obtain By combining ( 54) and ( 56), we see that F (t) is equal to a constant, precisely to F 0 := F (0) = F (0), and hence F (t) = F 0 exp(−αγt) .Since F (t) > 0 for every t ∈ [0, T ), from the equation we see in particular that the family of convex sets C(t) is strictly decreasing with respect to t. (ii) We recall that, for a convex body K, we have the inequalities (57) (the first one is the Saint-Venant inequality (19), while for the second one we refer e.g. to [18,Proposition 11]).It follows that C(t) is a nondegenerate convex body for every t, and hence that T = +∞.The strict convexity of C(t) follows from the strict positivity of its Gaussian curvature resulting from the equation.
In particular, from (61) we conclude that ψ ∞ = a γ F 0 , namely the cone variational measure of C ∞ is a constant multiple of H n−1 S n−1 .

Concluding remarks
Remark 12.In dimension n = 2 and for K smooth, the existence of a solution to the logarithmic Minkowski problem (13) easily follows from the results proved in [13,Section 6].Indeed, it is not restrictive to assume T (K) = 1 and to minimize over convex bodies with unit torsion.Let {L j } be a minimizing sequence.By arguing as in the proof of Lemma 6.2 in [13], it is possible to find a sequence of origin-symmetric parallelograms {P j }, with T (P j ) = 1 and orthogonal diagonals, such that P j ⊆ L j ⊆ 2P j .If {L j } is not bounded, we have that also {P j } is not bounded; moreover, by the assumption T (L j ) = 1, the monotonicity of torsion by inclusions, and Saint-Venant inequality (19), the volume of P j is bounded from below by a positive constant independent of j.Then we can apply Lemma 6.1 in [13] to infer that the sequence S 1 log h P j dV K is not bounded from above.Since by Hopf's boundary lemma |∇u K | 2 has a strictly positive minimum on ∂K, this implies that also the sequence S 1 log h L j dτ K is not bounded from above, contradicting the fact that {L j } is a minimizing sequence.Hence, {L j } remains bounded, and by Blaschke's selection theorem we may select a converging subsequence; its limit turns out to be an origin symmetric convex body with unit torsion which solves problem (13).

1 ψS n− 1 ψ
n (ξ) dH n−1 = ∞ (ξ) dH n−1 , by with equality if and only if L = B, where V B := V B /|B| denotes the normalized cone volume measure of B. Since B is a ball, we have V B = τ B .Then, by using (62) and the Saint-Venant inequality, we obtain * .