Small perturbations of polytopes

Motivated by first-order conditions for extremal bodies of geometric functionals, we study a functional analytic notion of infinitesimal perturbations of convex bodies and give a full characterization of the set of realizable perturbations if the perturbed body is a polytope. As an application, we derive a necessary condition for polytopal maximizers of the isotropic constant.


Introduction
In this article, a convex body is a non-empty compact convex subset of R n .We equip R n with the standard scalar product ⟨•, •⟩.The space of convex bodies in R n is denoted by K n and the subspace of full-dimensional convex bodies by K n n .In convex geometry, one often encounters functionals ϕ : A → R of the form (1.1) where A ⊂ K n , f 1 , . . ., f m are continuous functions R n → R and g : R m → R is differentiable.Examples of such functionals include the volume, components of the centroid, the moment of inertia, the difference between the left-hand and right-hand sides in the B-theorem, and the isotropic constant.If one is interested in extremal questions related to such functionals, it is natural to ask how ϕ changes if K is slightly "perturbed", i.e., replaced by another convex body that is close to K. Instead of considering a single perturbation, we consider a whole family of convex bodies, absolving us temporarily of the question what we mean exactly by "close".Let (K t ) t∈[0,1] be a one-parameter family of convex bodies in A for which the derivatives d dt Kt f i (x) dx t=0 exist.Then the derivative of ϕ "along" (K t ) t∈[0,1] is given by As an example, we consider the isotropic constant L K of a full-dimensional convex body K ∈ K n n , which is given by , where c K = 1 vol K K x dx.
Because the centroid c K modifies the domain of integration, the functional L 2n K is not of the form (1.1), but this turns out to be a minor technicality.Let A ⊂ K n n be the class of centered convex bodies, i.e., those K ∈ K n n with c K = 0. Defining ϕ : K n n → R by we clearly have ϕ(K) = L 2n K for K ∈ A. Let (K t ) t∈[0,1] be a one-parameter family in K n n (not necessarily in A).It was shown by Rademacher [16] that if K 0 is isotropic, i.e., it satisfies then the derivatives of t → ϕ(K t ) and t → L 2n Kt at t = 0 coincide and are given by if the derivative on the right-hand side exists.Using certain one-parameter families (K t ) t∈[−1 ,1] , which can intuitively be described as "hinging the facets", Rademacher used the first-order conditions that follow from (1.3) to derive the remarkable conclusion that a simplicial polytope P that maximizes K → L K must be a simplex [16].Rademacher's one-parameter family from [16].The gray facet hinges around the thickly drawn edge.
The present article is motivated by the question whether similar perturbation arguments might be used to derive further information about the extremal bodies of the isotropic constant (similar arguments are used in [17] and [11]).Phrased somewhat more generally, one might ask which derivatives "can appear" on the right-hand side of (1.3).Clearly, in order to make this question meaningful, some geometric restrictions on the family (K t ) t∈ [0,1] are needed to ensure that it actually describes a "local" perturbation.To make this notion precise, we return to the more general setting of the derivative given in (1.2).The exact geometric restriction that we impose on the family (K t ) t∈ [0,1] is that all derivatives of the form (1.2) exist, i.e., for all functionals ϕ of the form (1.1).This leads us to the following definition, where we denote by C(R n ) the vector space of continuous functions R n → R. Definition 1.1.Let (K t ) t∈[0,1] be a family of full-dimensional convex bodies.For every f ∈ C(R n ), we obtain a function ϕ f (t) : [0, 1] → R via We say that (K t ) t∈ [0,1] is weakly differentiable if ϕ f (t) is differentiable (from the right) at t = 0 for every f ∈ C(R n ).The linear map is called the weak derivative of (K t ) t∈ [0,1] .
Using an analogy from differential geometry, the weak derivative K ′ 0 can be thought of as a "tangent vector" at K 0 to the space K n n , which describes the directional derivative along the curve (K t ) t∈[0,1] at t = 0. Based on Definition 1.1, we are now able to state our problem more precisely.Question 1.2.Which linear maps C(R n ) → R can be realized (i.e., can appear) as weak derivatives of families (K t ) t∈[0,1] ?
A finite signed Borel measure on R n is a set function of the form α 1 µ 1 + α 2 µ 2 , where µ 1 , µ 2 are probability measures on the Borel σ-algebra and α 1 , α 2 ∈ R are arbitrary scalars.In the following, we will refer to such set functions simply as signed measures.Defining the support of a (signed) measure is somewhat complicated; for our purposes, it suffices to define that "µ is supported on A" means µ(B) = 0 for all measurable sets B ⊂ X \ A. Let X ⊂ R n be a Borel set.We denote the vector space of signed measures supported on X by M(X) and the cone of positive (signed) measures supported on X by M + (X).Moreover, for µ ∈ M(R n ), we use the notation µ| X to denote the restricted measure µ( • ∩ X).
Using the Riesz-Markov-Kakutani representation theorem, we will show in Section 2 that weak derivatives can be represented by signed measures on the boundary of K 0 .Formally, this statement reads as follows.
Proposition 1.3.A family (K t ) t∈[0,1] is weakly differentiable if and only if there exists a signed measure µ ∈ M(bd K 0 ) such that d dt ϕ f (t) Throughout the article, we will identify the weak derivative K ′ 0 with its representing measure µ ∈ M(bd K 0 ).Proposition 1.3 shows that the weak derivative from Definition 1.1 coincides with a notion of weak derivatives of probability measures due to Pflug [12].Such weak derivatives were studied for convex bodies by Weisshaupt in [24] and Weisshaupt and Pflug in [13].For a broader discussion of different notions of derivatives of set-valued maps, we refer to [8,Sect. 1].
The following terminology reflects the fact that in Question 1.2 the weak derivatives themselves are the objects of primary interest, rather than the families that realize them.Definition 1.4.We call a signed measure µ ∈ M(bd K) a small perturbation of K ∈ K n n if there is a weakly differentiable family The small perturbation µ is called reversible if −µ is also a small perturbation of K. We denote the set of small perturbations of K by W(K) and the set of reversible small perturbations of K by W ± (K).
Evidently, Definition 1.4 customizes the convenient but somewhat casual term "small perturbation" to our specific purpose of finding first-order conditions -in other contexts, it might be appropriate to consider higher-order representatives of weakly differentiable families, or to apply a completely different notion of "small perturbations".
In light of this terminology, Question 1.2 can be rephrased as follows: , what are the sets W(K) and W ± (K)?A signed measure ν ∈ M(X) is called absolutely continuous with respect to another signed measure µ ∈ M(X) if one of the following two equivalent conditions (see [10,Sect. 7.5]) holds: i) for every Borel set A ⊂ X with µ(A) = 0, we have ν(A) = 0; ii) there exists a density, i.e., a function f ∈ L 1 (X) with ν(A) = A f dµ for all Borel sets A ⊂ X.Let H n−1 ∈ M(bd K) be the (n − 1)-dimensional Hausdorff measure (which assigns to each Borel set A ⊂ bd K the corresponding surface area).For a function f : K → R, let µ f ∈ M(bd K) be the signed measure that is given by µ f (A) = A f dH n−1 for all Borel sets A ⊂ bd K.In [24], Weisshaupt showed that We will extend Weisshaupt's result in the special case that K is a polytope, i.e., the convex hull of a finite set V ⊂ R n .We denote the spaces of polytopes and full-dimensional polytopes in R n by P n and P n n , respectively.For a comprehensive overview of the combinatorial theory of polytopes, we refer to [26].
we drop the non-negativity requirement, i.e., a 1-convex function is simply a convex function C → R. Let P ∈ P n .For α ∈ (0, 1], we denote by M cvx α (P ) the space of all signed measures that are absolutely continuous with respect vol dim P ∈ M(P ) with an α-convex density f : relint P → R.
We are now ready to state our main result, which gives a full characterization of the sets W (P ) and W ± (P ) for P ∈ P n n .Here and in the following, Φ(P ) is the set of proper faces of P and Φ m (P ) is the set of faces of dimension m.Theorem 1.7.Let P ∈ P n n .The set of small perturbations of P is given by the direct sum This set is a convex cone.
Translated into the setting of the Reynolds transport theorem (see, e.g., [18]), Theorem 1.7 asserts that the velocity function that describes the evolution of bd K t has to be concave on the relative interior of each facet F ⊂ P .With regard to the lower-dimensional faces, Theorem 1.7 shows that the restriction of a small perturbation to a face F ⊂ P with dim F < n − 1 is a negative measure, which has a density f such that −f is (n − dim F ) −1convex.This conclusion resembles the reverse Brunn-Minkowski inequality for coconvex sets [9,20,7].
Theorem 1.7 leads to the following straightforward corollary.
Corollary 1.8.Let P ∈ P n n .The set of reversible small perturbations of P is given by W ± (P ) = This set is a vector space of dimension #Φ n−1 (P ) • n.
The rest of this text is organized as follows: In Section 2, we discuss some functional analytic preliminaries and show that weak derivatives can be represented by signed measures.The four subsequent sections are devoted to the proof Theorem 1.7.In Section 3, we construct polyhedral perturbations of a polytope, which are given by piecewise affine densities on the facets.In Section 4, we discuss the Wasserstein distance, which enables us to metrize weak convergence on compact sets.Using this tool, we show in Section 5 that the set of polyhedral perturbations is dense in the set that we claim to be W(P ).To complete the proof of our main result, we show in Section 6 that there are no other small perturbations of P .Finally, we discuss a necessary condition for polytopal maximizers of the isotropic constant in Section 7.

Weak derivatives and signed measures
In this section, we study some basic properties of weakly differentiable families.Using the Riesz-Markov-Kakutani representation theorem, we show that weak derivatives correspond to signed measures on the boundary of bd K 0 .
It is common to equip K n with the Hausdorff metric, which is defined by Another metric on K n is defined in terms of the symmetric difference K△L := K \L∪L\K.
It is called the symmetric difference metric and is given by It was shown in [22] that d H and d S induce the same topology on K n .The following lemma shows that a weakly differentiable family (K t ) t∈[0,1] is continuous at t = 0 with respect to both d H and d S .
(i) We have In particular, K t converges to K 0 in the Hausdorff metric as t → 0, which has the following consequences: (ii) For every ε > 0, there exists a t ε ∈ (0, 1] such that K t ⊂ K 0 + εB n for all t ≤ t ε , where B n denotes the centered Euclidean ball of radius 1. (iii) For every convex body L ⊂ int K 0 , there exists a t L ∈ (0, 1] such that L ⊂ K t for all t ≤ t L . Proof.We assume the negation of (i), which is equivalent to the statement that there exists a decreasing sequence (t i ) i∈N converging to zero with vol(K t i △K 0 ) ≥ 2 i • t i for all i ∈ N. Without loss of generality, let 0 ∈ int K.In the following, we will use the gauge function ∥ • ∥ K 0 : R n → R of K 0 , which is given by We set For every i ∈ N, there exists an r i ∈ (0, 1) with vol(M i ) ≥ 2 i−1 • t i .Clearly, the values r i can be chosen such that r i+1 < r i for all i ∈ N. The sequence (r i ) i∈N induces a partition of (0, 1) into half-open subintervals We define f : [0, 1] → [0, ∞) by setting f (r i ) := 1 i , f (0) := 0, f (1) := 1 and interpolating linearly on the intervals I i .Since f is continuous, the function g : R n → R given by g(x) := sgn(∥x∥ is continuous as well.By construction, we have g(x) This shows that the function ϕ g (t) from Definition 1.1 is not differentiable from the right at t = 0.The claim of (ii) is an immediate consequence of the definition of d H , for (iii) see [21,Lem. 1.8.18].□ In order to state the Riesz-Markov-Kakutani representation theorem, we review some preliminaries from functional analysis.
Let X ⊂ R n .We denote by C b (X) the Banach space of bounded continuous functions X → R, equipped with the norm ∥f ∥ ∞ := sup x∈X |f (x)|.Two subspaces of C b (X) are relevant for our purposes: i) The subspace of C b (X) that contains all functions f whose support is compact is denoted by C c (X).The normed space C c (x) is not complete.ii) The closure of C c (x) is denoted by C 0 (X) and contains all functions f that vanish at infinity in the following sense: for every ε > 0 there exists a compact set K ⊂ X with |f (x)| < ε for all x / ∈ K.The (topological) dual space of C 0 (X) is denoted by C 0 (X) * and consists of all continuous linear functionals C 0 (X) → R. A linear functional h : C 0 (X) → R is continuous if and only if its operator norm For X ⊂ R n , let µ ∈ M(X) be a signed measure.Hahn's decomposition theorem [10,Thm. 7.43] asserts that there exists a set X + ⊂ X such that i) µ(A) ≥ 0 for all A ⊂ X + ; and ii) µ(A) ≤ 0 for all A ⊂ X − := X \ X + .The partition (X + , X − ) is called a Hahn decomposition of X with respect to µ. Setting µ + := µ( • ∩ X + ) and µ − := −µ( • ∩ X − ), we have µ = µ + − µ − .The pair (µ + , µ − ) is called the Jordan decomposition of µ; it does not depend on which particular Hahn decomposition is chosen (see [10,Cor. 7.44]).The positive measure |µ| := µ + + µ − is called the total variation of µ.Definition 2.2.Let µ ∈ M(X) be a signed measure with Jordan decomposition (µ + , µ − ).The expression defines a norm on M(X) (see [10,Cor. 7.45]), which is called the total variation norm.
In the following, we will be mainly concerned with signed measures that are absolutely continuous with respect to the volume (Lebesgue measure) on some affine subspace of R n .For a Borel set A ⊂ R n and f ∈ L 1 (A), we use the notation [f ] M(A) to denote the signed measure on A that is absolutely continuous with respect to vol dim aff A with density f .If f is only defined on a subset of A, we use the trivial extension of f to A as a density.Moreover, if aff A = R n , we simply write [f ] M .For such signed measures, we have We now state a special case of the Riesz-Markov-Kakutani representation theorem; see for example [6,Thm. 2.26] or [1,Thm. 14.14].
Since C c (X) is dense in C 0 (X), a continuous linear functional C 0 (X) → R is uniquely determined by its values on C c (X).In particular, C c (X) separates points in C 0 (X) * .Remark 2.4.Theorem 2.3 is usually stated for more general topological spaces X, namely locally compact Hausdorff spaces.For arbitrary locally compact Hausdorff spaces, the map µ → I µ is not injective, unless we demand that the representing measure µ be regular, i.e., we require that for every Borel set A ⊂ X and ε > 0, there exist a compact set K and an open set U such that K ⊂ A ⊂ U and µ(U \ K) < ε [6, Def.2.21].In our setting, we can drop the requirement since by [4, (13.7.9)] and [4, (13.7.7)] every positive finite measure on R n is regular, and hence, by virtue of the Jordan decomposition, every signed measure on R n is regular.Definition 2.5.Let X ⊂ R n be open or closed and let (µ i ) i∈N be a sequence in M(X).We say that (µ i ) i∈N converges weakly to µ ∈ M(X) and write It follows from Theorem 2.3 that weak limits are unique.Remark 2.6.Two additional notions of convergence are obtained if we require that condition (2.1) holds for all f ∈ C c (X) or for all f ∈ C 0 (X), respectively.For the purposes of this article, the differences between these notions are not important since we can, considering small perturbations of a given convex body K ⊂ R n , restrict our attention to signed measures in M(K + B n ).Some consequences concerning the differentiability of parametric families of measures encountered in the case of non-compact support are discussed in [25, Ex. 2.12-Ex.2.15].For a general comparison of topologies and convergence on spaces of measures, including the notions of convergence induced by the functions in C b (X) and C c (X), respectively, we refer to [4,Sect. 13.20: Prob. 1,2].
Proof of Proposition 1.3.We only have to show that every weak derivative can be represented by a signed measure µ ∈ M(bd K 0 ).In light of Theorem 2.3, this amounts to showing that K ′ 0 can be written as where the last step follows from Lemma 2.1(ii).This shows that K ′ 0 can be written as We obtain that the trivial extension f given by is continuous, and, using Lemma 2.1(ii)-(iii), that [1 Kt − 1 K 0 ]f (x) dx = 0 holds for all t smaller than some t f > 0. Hence, we have

Polyhedral perturbations of polytopes
This section begins the constructive part of the proof of Theorem 1.7.The basic building blocks of our construction are polyhedral perturbations, which are described by densities on the boundary of a polytope P whose restrictions to relint F are piecewise affine and concave for each facet F ⊂ P .Using an elementary construction, we show that every polyhedral perturbation can be realized as a weak derivative.
An affine functional on an affine subspace H ⊂ R n is an affine function H → R.
for a finite family H of affine functionals aff(K) → R. Given a polyhedral concave function f : K → R, there exists a unique family H that provides an irredundant description of f in the sense that for each h ∈ H there is an Assuming that H is irredundant, the expression (3.1) can be read as describing a polyhedral function f : aff(K) → R. We call f the canonical extension of f .
The main result of this section reads as follows.
Proposition 3.2.Let P ∈ P n n .Moreover, let f : bd P → R be a measurable function with the property that the restriction f | relint F is concave and polyhedral for each facet F ⊂ P .Then we have We call a small perturbation of the form (3.2) a polyhedral perturbation and denote the set of polyhedral perturbations of P by W ◁ (P ).
The geometric idea behind the following construction is to choose a generic direction v ∈ S n−1 , introduce a redundant constraint for every affine functional in the description of f , and shift the corresponding hyperplanes in such a way that the velocities in direction v are proportional to the values of f at the corresponding boundary points of P .Construction 3.3.Let P and f be as in Proposition 3.2 and assume that P is given by We assume that this description is irredundant and denote the facet with outer normal vector u i by F i , i ∈ [m].Let v ∈ S n−1 be generic with respect to P in the sense that ⟨u i , v⟩ ̸ = 0 for i ∈ [m].We set Because v is generic, we have

Denoting the orthogonal projection onto v
Then P can be written as We denote this bijection by π i and define a function , By construction, u( • , t) and −ℓ( • , t) are concave and polyhedral for all t ∈ [0, 1].Therefore, ) is either a polytope or the empty set for t ∈ [0, 1].We obtain a family of full-dimensional polytopes (P t ) t∈[0,1] that satisfies P 0 = P by setting Remark 3.4.For later reference we note that Construction 3.3 has the following property: For any t ∈ [0, 1], the volume of P t \ P is given by and hence, using (3.4), satisfies the bound Example 3.5.Let P ⊂ R 3 be a simplicial polytope and let v ∈ S 2 be generic with respect to P .Moreover, let F ⊂ P be a facet.For the sake of simplicity, we assume that  1] corresponds to the operation of stacking a pyramid Q onto the facet F , where the apex of Q moves outwards with constant velocity.Recalling Corollary 1.8, we note that (i) and (ii) describe reversible small perturbations, whereas (iii) is not reversible.Proposition 3.2 follows from the following claim.
Claim 3.6.The family (P t ) t∈[0,1] from Construction 3.3 is weakly differentiable with Before we proceed to show Claim 3.6, we prove some auxiliary statements.
Lemma 3.7.Let P and f be as in Proposition 3.2 and f i , i ∈ [m], be as in Construction 3.3.We define two functions v ⊥ → R by The functions f u and f ℓ and the functions u, ℓ from Construction 3.3 have the following properties: (i) For almost all x ∈ π v ⊥ (P ) there is an Proof.For (i), we first observe that holds for all t ≥ 0 and we can choose ε + x > 0 arbitrarily.Otherwise, the choice ε In both cases, we have We are now ready to verify that the family of polytopes (P t ) t∈[0,1] from Construction 3.3 realizes the small perturbation given by the density f .Proof of Claim 3.6.Let g ∈ C(R n ) be arbitrary.Setting and our goal is to show that lim t→0 + G(t) = g dµ.Without loss of generality, we can assume that v is the n-th standard basis vector e n and that g is bounded (as the claim only depends on the values of g on, say, 3), we observe that , where We first consider G 2 (t).On the set π v ⊥ (P ) \ π v ⊥ (P t ) we have u(x, t) < ℓ(x, t), which implies, using Lemma 3.7(ii), It follows that, for all (x, t) is a dominating integrable function and we can apply the Lebesgue dominated convergence theorem to obtain Turning to G 1 (t), we observe that it can be rewritten as Again using Lemma 3.7(ii), we have and a similar bound for ) is again a dominating integrable function.Applying Lemma 3.7(i), the Lebesgue dominated convergence theorem and the fundamental theorem of calculus, we obtain

The Wasserstein distance
We say that a sequence The goal of this section is to metrize weak convergence of bounded sequences in M(X) for a compact set X ⊂ R n .As a byproduct of the metrization, we will obtain the following result.Remark 4.2.Let X ⊂ R n be compact.Then the ∥ • ∥ TV -unit ball of M(X), which we denote by B M(X) , is compact and metrizable (and hence also sequentially compact) with respect to the vague topology, which is induced by the family of all functions g ∈ C c (R n ) via µ → g dµ.By Remark 2.6 and the compactness of X, the vague topology coincides with the topology induced by all functions g ∈ C b (R n ) via µ → g dµ.We consider a bounded sequence (µ i ) i∈N in M(X).Then there exists a signed measure µ ∈ M(X) and a subsequence (µ This is a direct consequence of [4, (13.4.1)] and [4, (13.4.2)(ii)] adapted to real measures (see [4,Sect. 13.2]), using that B M (X) is for compact X closed in the vague topology on the space of all real measures on R n .That the real measures in the sense of [4,Sect. 13.2] include the signed measures in the sense of this article follows from [4, (13.7.9)] and [4, (13.3.6)].
Instead of using the mentioned metrizability result "off the shelf", this section uses tools from optimal transport theory to develop a metrization of weak convergence on the unit ball of M(X).The chosen metrization is based on the Wasserstein norm from [15].In addition to proving Proposition 4.1, this metrization provides a useful tool for analyzing weak convergence of concrete sequences in M(R n ).
We first recall some basic notions from optimal transport theory.For a comprehensive introduction to this subject, we refer to [23] , where π i denotes the projection onto the i-th coordinate and (π i ) * µ is the pushforward of µ, given by (4.1) ) for all Borel sets A ⊂ X i .More generally, we define the pushforward f * µ ∈ M(Y ) for any measurable map f : X → Y and a signed measure µ ∈ M(X) as in (4.1).
The generalized Wasserstein distance of µ and ν is defined by where the infimum is taken over all μ, ν As shown in [14, Prop.1], the infimum in (4.3) is attained and does not change if we impose the additional condition that μ(A) ≤ µ(A) and ν(A) ≤ ν(A) holds for all Borel sets A ⊂ R n .
We say that a sequence The significance of the generalized Wasserstein distance lies in the fact that it metrizes weak convergence of tight sequences of positive measures.The following result is taken from [14,Thm. 3].
We now turn to signed measures.Definition 4.5.Let µ ∈ M(X) with Jordan decomposition (µ + , µ − ).Following [15, Def.17], we define the Wasserstein norm by It is shown in [15,Prop. 21] that ∥ • ∥ W is indeed a norm, and it is easy to see that ∥µ∥ W ≤ ∥µ∥ TV holds for all µ ∈ M(X).The following sequential compactness result will be used extensively below, especially in Section 6.
The utility of ∥ • ∥ W for our purposes lies in the following fact.
Proposition 4.6.Let X ⊂ R n be compact.The Wasserstein norm metrizes weak convergence of bounded sequences in M(X), i.e., for a bounded sequence (µ i ) i∈N in M(X) we have Proof.We begin with the forward implication.Let f ∈ C(X) and µ ∈ M(X) with ∥µ∥ W < ε for some ε > 0. Then there exists a transference plan τ ∈ M + (X × X) with τ (X × X) ≤ µ + (X) and Since ∥µ∥ W < ε, we have X×X ∥x − y∥ 2 dτ (x, y) < ε and hence Writing To bound the integral on the right-hand side, we first observe that Since X is compact, f is uniformly continuous.Therefore, the function satisfies lim t→0 + ω(t) = ω(0) = 0. Recalling (4.4), we have Putting the estimates together, we have which shows the forward implication.
For the other implication, we assume that µ i w → 0. By Remark 4.2, there exists a subsequence of µ k(i) whose Jordan decomposition satisfies µ k(i) The triangle inequality yields Remark 4.7.For x ∈ [−1, 1], let δ x be the Dirac measure supported on {x}.The sequence ( √ i(δ 1/i −δ −1/i )) i∈N shows that the forward implication in Proposition 4.6 fails without the assumption that (µ i ) i∈N is bounded.On the other hand, if we assume that the sequence (µ i ) i∈N is weakly convergent, then its boundedness follows from the uniform boundedness principle [3,Thm. 14.1].
We conclude this section with the proof of the result from its beginning.
Proof of Proposition 4.1.Let v ∈ S n−1 be generic with respect to P .For each i ∈ N, let (P i t ) t∈[0,1] be the family from Construction 3.3 that realizes µ i ∈ W ◁ (P ).We fix i ∈ N. By Remark 3.4, we have Combining these two facts, we obtain that lim sup Hence, by Lemma 2.1(i) and Proposition 4.6, there exists an r i > 0 such that Clearly, the values r i can be chosen such that r i+1 ≤ 1 i for all i ∈ N.For t ∈ (0, 1], we set i(t) := max{i ∈ N | r i ≥ t} and P t := P i(t) t .Finally, we set P 0 := P .By Remark 4.7, we have sup i∈N ∥µ i ∥ TV ≤ C for some C > 0, which implies that lim sup Using Proposition 4.6 in combination with the triangle inequality for ∥ • ∥ W , it follows that (P t ) t∈[0,1] realizes the small perturbation µ. □

Weak limits of polyhedral perturbations
This section completes the constructive part of the proof of Theorem 1.7.Using an approximation by polyhedral perturbations, we will show the first inclusion of Theorem 1.7, which reads as follows.
Proposition 5.1.Let P ∈ P n n .The set of small perturbations of P satisfies The following theorem is a classical result in convex analysis [19,Thm. 10.8].

Theorem 5.2. Let C ⊂ R n be relatively open and convex and let
and the convergence is uniform on each closed bounded subset of C.
A convex function f defined on a convex set K ⊂ R n is called polyhedral if −f is polyhedral.The first step of our approximation argument is contained in the following lemma.
Lemma 5.3.Let P ∈ P n .For α ∈ (0, 1], let f : relint P → R be an integrable α-convex function.For every ε > 0, there exists a polyhedral convex function g : relint P → R with In particular, we have ∥[f − g Proof.Let ε > 0. Without loss of generality, we assume that P is full-dimensional.By assumption, f α is convex.Let (x i ) i∈N be a sequence whose image is dense in int P .For every i ∈ N, let h i : int P → R be an affine functional with h i (x i ) = f α (x i ) and h i ≤ f α .We define g i by The function g 1/α 1 : int P → R is bounded and hence integrable.Therefore, there exists a compact set Q ⊂ int P with for all i ∈ N.
Since (g i ) i∈N converges pointwise to f α on the dense set {x i } i∈N ⊂ P , Theorem 5.2 asserts that (g i ) i∈N converges uniformly to f α on the compact set Q. Since the sequence (g i ) i∈N is monotonically increasing and bounded from above by the continuous function f α , there exists an interval [a, b] (if α < 1, we can choose a ≥ 0) with The following lemma allows us to approximate a measure with an (n − dim F ) −1 -convex density on a lower-dimensional face F of P by an iterative argument over the face lattice.Lemma 5.4.Let P ∈ P n and F ⊂ P a facet.For m ≥ 2, let f : relint F → R be a 1  m -convex integrable function.For every ε > 0, there exists a 1 m−1 -convex integrable function g : relint Proof.Let ε > 0. By Lemma 5.3, there exists a polyhedral convex function h on relint F such that ∥[f − h m ] M(F ) ∥ TV < ε.We denote the orthogonal projection onto aff F by π aff F and the canonical extension of h by h.For δ > 0, we define h δ : relint P → R by Since x → h(π aff F (x)), x → − dist(x, aff F ) and x → 0 are convex functions on relint P , h δ is convex as well.Therefore, the function It remains to show that δ > 0 can be chosen such that W (µ δ , ν) becomes arbitrarily small.We first choose a polytope G ⊂ relint F such that ν(F \ G) < ε.Let u F ∈ S n−1 be the outer normal vector of F .Since G ⊂ relint F , there exists a δ 1 > 0 such that conv{G, G − δ 1 • u F } ⊂ P .We decompose P into three parts: We first consider P 1 .If δ ≤ δ 1 , then Fubini's theorem implies that a density of (π aff F ) * (µ δ | P 1 ) is given by which shows that (π aff F ) * (µ δ | P 1 ) = ν| G .Using the transference plan τ induced by the function π aff F , i.e., setting τ := (id, π aff F ) * (µ δ | P 1 ), we obtain that the Wasserstein distance of µ δ | P 1 and ν| G is bounded by In principle, the densities of (π aff F ) * (µ δ | P 2 ) and (π aff F ) * (µ δ | P 3 ) can be computed similarly, but for y ∈ π aff F (P ) \ G the segment [y, y − δ • h(y) • u F ] is not necessarily contained in P .Therefore, we have the upper bounds µ δ (P 2 ) ≤ ν(F \ G) and Since h is bounded on π aff F (P ), there is a δ 2 > 0 such that δ ≤ δ 2 implies µ δ (P 3 ) < ε.
We now combine the bounds for P 1 , P 2 and P 3 .If then by the estimates above we have By the triangle inequality, it follows that With regard to the total variation norm, we observe that We are now ready to prove the main result of this section.
Proof of Proposition 5.1.Let µ ∈ F ∈Φ(P ) −M cvx (n−dim F ) −1 (F ).We consider a face F ∈ Φ(P ) and define We repeat this construction for every F ∈ Φ(P ) and set Performing this procedure for every i ∈ N, we obtain a sequence (µ i ) i∈N in W ◁ (P ) which satisfies as well as By Proposition 4.6 and Proposition 4.1, it follows that µ ∈ W(P ).□

Weak limits of contrast sequences
In this section, we complete the proof of Theorem 1.7.It remains to show the second inclusion, i.e., that there are no other small perturbations than the ones that we constructed in the preceding sections.Definition 6.1.Let P ⊂ R n be a full-dimensional polytope.A sequence of signed measures Clearly, the sequence [(β i , K i )] i∈N is uniquely determined by (µ i ) i∈N ; we call it the component sequence of (µ i ) i∈N .A P -contrast sequence (µ i ) i∈N is called admissible with limit µ ∈ M(R n ) if (i) (µ i ) i∈N converges weakly to µ, (ii) (β i ) i∈N diverges to infinity, and (iii) there exists a compact set C ⊂ R n such that µ i is supported on C for all i ∈ N. We note in passing that condition (iii) is logically redundant and merely included for convenience.
Evidently, Definition 6.1 is closely related to weak differentiability: If (K t ) t∈[0,1] is weakly differentiable, then [(i, K 1/i )] i∈N is the component sequence of a P -contrast sequence.Translated into the terminology of Definition 6.1, the problem we face is to characterize limits of P -contrast sequences.As we will discuss below, parts of our proof are inspired by the elegant proof of the reverse Brunn-Minkowski inequality for coconvex bodies by Fillastre [7].
We begin with some basic properties of contrast sequences.Lemma 6.2.Let P ∈ P n n and (µ i ) i∈N an admissible P -contrast sequence with limit µ. (i) The sequence (µ i ) i∈N is bounded.(ii) The sequence (K i ) i∈N converges to P with respect to d H . (iii) The limit µ is supported on bd P .
Proof.As discussed in Remark 4.7, (i) follows from the uniform boundedness principle, using that µ i is supported on a fixed compact set C for all i ∈ N. Since β i diverges to infinity, the boundedness of (µ i ) i∈N implies that d S (K i , P ) converges to zero.Using the fact that d H and d S induce the same topology on K n , this shows (ii).Finally, (iii) follows from (ii) by a similar argument as in the proof of Proposition 1.

□
The following lemma is a partial version of the continuous mapping theorem for signed measures.Lemma 6.3.Let (µ i ) i∈N be a weakly convergent sequence in M(R n ) and π : R n → R n a continuous map.We have Our basic strategy for characterizing limits of P -contrast sequences is to consider the signed measures locally, i.e., on appropriately chosen compact subsets of R n .Lemma 6.4.
We note that the trivial extension f : R n → R is an element of C(R n ).Therefore, we have After restricting the P -contrast sequence under consideration to an appropriately chosen compact subset of R n , we want to show that the weak limit of the restricted sequence is absolutely continuous with a concave density.For this, we need the following lemma.Lemma 6.5.Let P, Q ∈ P n with Q ⊂ relint P and m := dim P = dim Q.Let (f i ) i∈N be a sequence of concave functions P → R that is bounded in L 1 (P ), i.e., we have Proof.We start with (i).Setting T := 1 2 (P + Q), we have Q ⊂ relint T ⊂ T ⊂ relint P .We first show that {f i | T } i∈N is uniformly bounded from below.For an arbitrary z ∈ P , let S := S n−1 ∩ (aff P − z).We define a function Because T ⊂ relint P , the set {x ∈ P | ⟨u, x⟩ ≤ ⟨u, y⟩} is an m-dimensional polytope for all (y, u) ∈ T × S. Therefore, we have h(y, u) > 0 for all (y, u) ∈ T × S. Since T × S is compact and h is continuous, it follows that Let i ∈ N and x ∈ T .If f i (x) ≥ 0, there is nothing to show, so we assume that f i (x) < 0. Because f i is concave, the superlevel sets of f i are convex, and hence there exists a u ∈ S such that f i (y) ≤ f i (x) for all y ∈ {x ∈ P | ⟨u, x⟩ ≤ ⟨u, y⟩}.This leads to the estimate To complete the proof of (i), we now show that {f i | Q } i∈N is uniformly bounded from above.Let i ∈ N and x ∈ Q.We distinguish two cases: a) If f i (y) ≥ 0 for all y ∈ Q, then by the concavity of f i , the (m + 1)-dimensional pyramid is contained in the subgraph of f i .Therefore, we have which yields a uniform upper bound on f i (x).
b) We assume there exists a v ∈ Q with f i (v) < 0. We define a map by stipulating that b(x, y) is the unique intersection point of relbd T with the ray that emanates from x and goes through y.Because Q ⊂ relint T , we have , then f i (v) < 0 yields the desired upper bound, so we assume that f i (x) > f i (v).Then, by the concavity of f i , we have and hence Combining both cases, we obtain that To show (ii), we construct the subsequence (f As the first step towards a characterization of limits of P -contrast sequences, we determine their behavior on the facets of P .Lemma 6.6.Let P ⊂ R n be a full-dimensional polytope and (µ i ) i∈N an admissible Pcontrast sequence with limit µ.Let F ⊂ P be a facet and G ⊂ relint F a polytope with dim G = dim F .Then µ| relint G is absolutely continuous with respect to vol n−1 ∈ M(F ) with a concave density f : relint G → R.
Proof.Let u F ∈ S n−1 be the unit outer normal vector of F .Setting G := Let (β i , K i ) i∈N be the component sequence of (µ i ) i∈N .By Lemma 6.2(ii), we can assume without loss of generality that G − αu F ⊂ K i for all i ∈ N. We define two polytopes By Remark 4.2, there exists a subsequence (µ Arguing as in the proof of Proposition 1.3, we obtain that ν is supported on G. Using Lemma 6.3 and Lemma 6.4, we have Because we assumed that G − αu F ⊂ K i for all i ∈ N, every g i is well-defined and bounded from below by αβ i .Moreover, the convexity of K i implies that g i is concave.It follows from Fubini's theorem that g i | G is a density of (π aff F ) * (µ i | Q ).By Lemma 6.2(i), there exists a constant C > 0 such that Hence, Lemma 6.5(ii) implies that we can find a subsequence . By the uniqueness of weak limits, it follows that We now extend the assertion of Lemma 6.6 to whole facets of P .Proposition 6.7.Let P ⊂ R n be a full-dimensional polytope, F ⊂ P a facet and (µ i ) i∈N an admissible P -contrast sequence with limit µ.Then µ| relint F is absolutely continuous with respect to vol n−1 ∈ M(F ) with a concave density f : relint F → R.
be the concave density of µ| relint F i whose existence was shown in Lemma 6.6.Let j, k ∈ N with j < k.Then f k and f j agree on relint F j , since both densities are continuous and describe the same signed measure on relint F j .Therefore, the function f : relint In the second step, we determine the behavior of limits of P -contrast sequences on lower-dimensional faces of P .The following characterization of weak convergence is part of the well-known Portemanteau theorem [10,Thm. 13.16].Theorem 6.8 (Portemanteau).Let X be a metric space and µ, µ 1 , µ 2 , . . .
We use Theorem 6.8 to show that the restriction of a limit of a P -contrast sequence to a face F ⊂ P with dim F < n − 1 is a negative measure.Lemma 6.9.Let P ⊂ R n be a full-dimensional polytope and (µ i ) i∈N an admissible Pcontrast sequence with limit µ.Let i.e., S is the union of all proper faces of F that are not facets.Then there exists a restricted subsequence (µ k(i) | P ) i∈N in M − (P ) := −M + (P ) that converges weakly to a signed measure ν with ν| S = µ| S .In particular, we have µ(A) ≤ 0 for all Borel sets A ⊂ S.
We decompose µ i into the sum By Definition 6.1(iii), there exists a compact set C ⊂ R n such that µ i is supported on C for all i ∈ N. Hence, Remark 4.2 implies that there exists a subsequence (µ k(i) ) i∈N such that every part of the decomposition converges weakly, i.e., there exist signed measures Let (β i , K i ) i∈N be the component sequence of (µ i ) i∈N .By Lemma 6.3, we know that its outer normal vector and z ∈ relint F .We define Let α > 0 be such that F 2 3 − αu F ⊂ int P .By Lemma 6.2(ii), there exists an N ∈ N such that (6.1) For i ∈ N, we define g i : Because K i is convex, g i is a concave function F 2 → R ∪ {−∞} for all i ∈ N.Moreover, by (6.1), the restriction is real-valued for all i ≥ N .Using Lemma 6.5(i), we obtain that there exist constants C ℓ , C u ∈ R such that Let ε > 0 and let U ⊂ aff F be (relatively) open with relbd F ⊂ U and vol n−1 (U ) < ε.By Theorem 6.8, we have Since ε > 0 was arbitrary, this implies ν 1 (S) = ν 1 (relbd F ) = 0.
To complete the proof, we observe that (µ i | P + m j=2 µ i | Q j ) i∈N is again a P -contrast sequence, and we can iterate the argument above to obtain that ν j (S) = 0 for j ∈ [m].□ Remark 6.10.Let P , (µ i ) i∈N , µ and S be as in Lemma 6.9.Moreover, let F ⊂ P be an m-face for 0 ≤ m < n − 1.Since F ⊂ S, Lemma 6.9 implies that there exists a restricted subsequence (µ k(i) | P ) i∈N that converges weakly to a signed measure ν with ν| F = µ| F .Therefore, for the purpose of determining µ| F , we can replace (µ i ) i∈N by (µ k(i) | P ) i∈N .
The localization argument for the lower-dimensional faces of P is somewhat more involved than the corresponding argument for the facets.Instead of restricting the sequence (µ i ) i∈N to a single compact set Q ⊂ R n , we consider a whole sequence of compact sets.Lemma 6.11.Let (µ i ) i∈N be a sequence in Proof.Applying Remark 4.2 iteratively, we obtain a subsequence (µ k(i) ) i∈N such that (µ k(i) | Q j ) i∈N converges weakly to a signed measure ν j for all j ∈ N. By Lemma 6.4, we have By the continuity of measures from above, we have For this, we observe that □ The following fact is a straightforward consequence of Fubini's theorem, we state it as a lemma for later reference.Lemma 6.12.Let H ⊂ R n be an affine subspace and π : R n → H a projection, i.e., an affine map with π| H = id H . Then there exists a constant γ(π) ∈ (0, 1] such that for all measurable sets A ⊂ R n . We now combine the insights of the previous lemmas into a result that allows us to identify the densities of limits of P -contrast sequences on lower-dimensional faces of P .Lemma 6.13.Let P ⊂ R n be a full-dimensional polytope and (µ i ) i∈N an admissible P -contrast sequence with limit µ and component sequence (β i , K i ) i∈N .Let F ⊂ P be an m-face for 0 ≤ m < n − 1 and G ⊂ relint F a polytope with dim G = dim F .Let π : R n → aff F be a projection.Moreover, let (Q j ) j∈N be a sequence of compact sets with For i, j ∈ N, let g j i : relint F → (−∞, 0] be given by , where γ(π) is the constant from Lemma 6.12.Finally, we assume that for every j ∈ N, the sequence (g j i ) i∈N converges uniformly on relint F to a function g j ∞ : relint F → R. Then the sequence of limits (g j ∞ ) j∈N converges pointwise to a function g * : relint F → (−∞, 0] and we have Proof.By Remark 6.10, we can assume without loss of generality that K i ⊂ P for all i ∈ N. Lemma 6.3 and Lemma 6.11 imply that there exists a subsequence (µ k(i) ) i∈N such that It follows from Lemma 6.12 that g j k(i) is a density of π * (µ k(i) | Q j ).By assumption, the sequence ([g j k(i) ] M(F ) ) i∈N converges to [g j ∞ ] M(F ) in the total variation norm and hence also weakly.It remains to show that wlim j→∞ [g Since the sequence (Q j ) j∈N is decreasing, the sequence (g j ∞ ) j∈N satisfies g j ∞ ≤ g j+1 ∞ ≤ 0 for all j ∈ N and hence converges pointwise to a function g * : relint F → (−∞, 0].By the monotone convergence theorem [5,Thm. 4.3.2], it follows that To show that limits of P -contrast sequences have concave densities on lower-dimensional faces of P , we make use of the following sharpened version of the Brunn-Minkowski inequality for convex sets with identical projections.This result is classical, see for example [2,Ch. 50 We now prove an analogous result to Lemma 6.6 for faces F ⊂ P with dim F < n − 1. Lemma 6.15.Let P ⊂ R n be a full-dimensional polytope and (µ i ) i∈N an admissible Pcontrast sequence with limit µ.Let F ⊂ P be an m-face for 0 ≤ m < n − 1.Then µ| relint F is absolutely continuous with respect to vol m ∈ M(F ) with a concave density The conclusion in Lemma 6.15 is weaker than the claim that −f (n−dim F ) −1 is convex, which is asserted in Theorem 1.7.We will strengthen this conclusion below.
Proof of Lemma 6.15.Let (β i , K i ) i∈N be the component sequence of (µ i ) i∈N .Again, by Remark 6.10, we can assume without loss of generality that K i ⊂ P for all i ∈ N. Let G ⊂ relint F be a polytope with dim G = dim F and set G j := j j+1 G + 1 j+1 F for j ∈ N. Let u 1 , . . ., u ℓ ∈ S n−1 be the outer normal vectors of the facets F 1 , . . ., F ℓ that contain F .For every v ∈ vert(G 1 ), there is an By Lemma 6.2(ii), we can assume without loss of generality that (6.2) By construction, the sequence (Q j ) j∈N satisfies the condition Q j+1 ⊂ int Q j from Lemma 6.13; the sequence ( Qj ) j∈N will allow us to apply Lemma 6.5.In the following, we will use Lemma 6.13 to show that µ| G is absolutely continuous with respect to vol m ∈ M(G) with a concave density relint G → (−∞, 0].The claim of the lemma can then be deduced by an argument as in the proof of Proposition 6.7.
For i, j ∈ N, let g j i : relint G 1 → R be given by We will first show that g j i is concave if i and j are sufficiently large.For x ∈ relint G 1 , let L j (x) := Qj ∩ π −1 aff F (x).We note that by (6.2) and ( 6.3) the definition of L j (x) implies that L j (x) ∩ K i ⊃ x − w 2 + 1 j B and hence (6.4) We consider the (n − m)-dimensional convex body L j (x) ∩ P for a given x ∈ relint G 1 .Let P F ⊃ P be the polyhedron that results from P by deleting all facet-defining inequalities of P that are not active at relative interior points of F , i.e., P F is bounded by the facetdefining hyperplanes aff F 1 , . . ., aff F ℓ .Then there exists an N ′ ≥ N such that (6.5) L j (x) ∩ P = L j (x) ∩ P F for all j ≥ N ′ .
In the following, we assume that i, j and hence, using (6.5), . By the convexity of K i , we have In combination with (6.7) and (6.6), this shows that g j i is concave for i, j ≥ N ′ .We fix j ≥ max{N ′ , 2}.By Lemma 6.2(i), the sequence (g j i ) i∈N is bounded in the space L 1 (G 1 ), and Lemma 6.5(ii) implies that there exists a subsequence (g j k(i) ) i∈N that converges uniformly on G j to a concave function g j ∞ : G j → (−∞, 0].Using Lemma 6.5(ii) iteratively together with the usual diagonal argument, we obtain a strictly increasing function k : N → N such that (g j k(i) ) i∈N converges uniformly on G j to g j ∞ for every j ≥ max{N ′ , 2}.With regard to the assumptions of Lemma 6.13, we observe that the sequence (Q j ) j∈N satisfies Q j ⊂ int Q j−1 for j ∈ N and bd P ∩ j∈N Q j = G, and we have ∞ is concave for every j ≥ N ′ , the claim follows.□ Our strategy for strengthening Lemma 6.15 is inspired by Fillastre's proof of the reverse Brunn-Minkowski inequality for coconvex bodies.The main idea of this proof is contained in the following lemma, which is taken from [7,Lem. 4].Lemma 6.16.Let C ⊂ R n be a convex cone and f : C → R a convex function that is positively homogeneous of degree m, i.e., f (λx) = λ m f (x) for all x ∈ C \ {0} and λ ≥ 0. Then f is 1 m -convex.To prove the reverse Brunn-Minkowski inequality, Fillastre first shows that the volume functional is convex on a given cone of coconvex bodies, and then uses the homogeneity of the volume to deduce the desired 1 n -convexity.Translated into our setting, this leads us to investigate how limits of P -contrast sequences behave under suitable projective transformations (in the sense of projective geometry).
Let (β i , K i ) i∈N be the component sequence of (µ i ) i∈N .Without loss of generality, we assume that K + i := K i ∩ H + ⊂ P + holds for all i ∈ N and that the restricted sequence (µ i | P + ) converges weakly.By a change of variables, we have for every f ∈ C(R n ).Since the function (f • τ ) • |det τ | is continuous on H + , it follows that (β i , τ (K + i )) i∈N is the component sequence of an admissible τ (P + )-contrast sequence (ν i ) i∈N with limit ν.
Applying Lemma 6.5(ii) iteratively as in the proof of Lemma 6.15, we obtain that there exists an N ′ ∈ N and a strictly increasing function k : N → N such that (g j k(i) ) i∈N converges uniformly on relint G j for all j ≥ N ′ .Since h −n+m is bounded on relint F + , it follows that (g j k(i) ) i∈N converges uniformly on τ (relint G j ).Applying Lemma 6.13 to the admissible τ (P + )-contrast sequence (ν i ) i∈N , we obtain that the pointwise limit g * (z) := lim j→∞ lim i→∞ gj i (z) exists and satisfies ν| τ (G) = [g * ] M (τ (F + )) .
Because ν| τ (relint F + ) has at most one continuous density and g * is continuous, Lemma 6.15 implies that g * | [r 1 x,r 2 y] is concave.Finally, recalling the definition of f , we observe that (6.8) implies g * | [r 1 x,r 2 y] = −γ(π) • f | [r 1 x,r 2 y] , which completes the proof.□ The proof of our main result is now merely a matter of combining the previous results.
Proof of Theorem 1.7.In light of Proposition 5.1, it remains to show that every µ ∈ W(P ) can be written as a direct sum as in (1.4).Let (K t ) t∈[0,1] be a family that realizes µ.Then [(i, K 1 i )] i∈N is the component sequence of a P -contrast sequence, and the claim follows from Proposition 6.7 and Proposition 6.17.The claim that W(P ) is a convex cone follows from the fact that W(P ) is the sequential closure of W ◁ (P ), which is clearly a convex cone.□ Remark 6.18.Theorem 1.7 implies that, for n ∈ N, the sum of two 1 n -convex functions is again 1 n -convex.This fact can also be deduced from the Hölder inequality.

A condition for polytopal maximizers of the isotropic constant
In this section, we use Theorem 1.7 to derive a necessary condition for polytopal maximizers of the isotropic constant and other geometric functionals.
Let ϕ : K n n → R be a functional of the form (1.1) and P ∈ P n n .We define the function h ϕ,P : R n → R by If P is a local maximizer of ϕ, then it follows from Theorem 1.7 that (7.1) h ϕ,P dµ ≤ 0 holds for all µ ∈ W(P ).In particular, (7.1) holds with equality for all µ ∈ W ± (P ).This leads us to the following definition.
Definition 7.1.Let ϕ be as above.A polytope P ∈ P n n is called perturbation stable (with respect to ϕ) if W(P ) is contained in the polar cone of {h ϕ,P }, i.e., if (7.1) holds for all µ ∈ W(P ).The polytope P is called weakly perturbation stable if W ± (P ) is contained in the annihilator of {h ϕ,P }, i.e., if (7.1) holds for all µ ∈ W ± (P ).
For F ∈ Φ(P ), we denote the restriction of the Lebesgue measure on aff F to F by vol F and set vol Φ := F ∈Φ(P ) vol F ∈ M + (bd P ).
Since W ◁ (P ) is dense in W(P ) and every element of W ◁ (P ) has a square-integrable density, a polytope P is perturbation stable if and only if W 2 (P ) is contained in the polar cone of {h ϕ,P }.This fact entails the following result.Proposition 7.2.Let P ∈ P n n and let ϕ and h ϕ,P be as above.We denote the metric projection onto the closed convex cone W 2 (P ) in the Hilbert space L 2 (bd P, vol Φ ) by π W .If P is a local maximizer of ϕ, then π W (h ϕ,P ) = 0 ∈ L 2 (bd P, vol Φ ).
Proof.The claim that W 2 (P ) is closed follows from the fact that L 2 -convergence implies L 1 -convergence (because vol Φ is finite), and the L 1 -norm coincides with the total variation norm.If there exists an f ∈ W 2 (P ) with ∥h ϕ,P − f ∥ L 2 < ∥h ϕ,P ∥ L 2 , then −2⟨h ϕ,P , f ⟩ < −2⟨h ϕ,P , f ⟩ + ∥f ∥ 2 L 2 < 0, and f is a witness for the fact that P is not perturbation stable.□ The necessary condition from Proposition 7.2 resembles the problem of convex regression in statistics.Since W 2 (P ) is a direct sum, the condition can be restated as follows: For each face F ⊂ P , the zero function is the unique minimizer of the risk minimization problem min f ∈F E[(−f (X) − h ϕ,P (X)) 2 ], where X is a random vector that is uniformly distributed on relint F and F denotes the class of square-integrable (n − dim F ) −1 -convex functions on relint F .
Rephrased for the isotropic constant, Proposition 7.2 reads as follows.

(a) unit cube C 3 3 Figure 2 . 1 t [ 1
Figure 2. Proposition 1.3 asserts that the family (K t ) t∈[0,1] is weakly differentiable if and only if the family of signed measures with densities 1 t [1 Kt − 1 K 0 ], t ∈ [0, 1], converges weakly (see Definition 2.5 below) to a signed measure on the boundary of K 0 as t → 0 + .

Figure 3 .
Figure 3.An illustration of Construction 3.3.Subfigure (A) refers to the functions f u and f ℓ introduced in Lemma 3.7 below.The sets P 0.1 \ P and P 0.2 \ P are shaded in cyan, whereas the sets P \ P 0.1 and P \ P 0.2 are shaded in orange.
3 with the following functions f , we obtain various basic perturbations of P .(i) Shifting F : If f = 1 F , then the family (P t ) t∈[0,1] corresponds to a continuous outward parallel shift of the facet F with constant velocity.(ii) Hinging F : Let E ⊂ F be an edge and set f := 1 F • dist( • , E).Then the family (P t ) t∈[0,1] corresponds to hinging the facet F outward around the edge E, where the dihedral angle between F and the hinged facet Ft at time t ∈ [0, 1] is equal to arctan(t).(iii) Stacking a pyramid onto F : If

Proposition 4 . 1 .
Let P ∈ P n n .The sequential closure of the set W ◁ (P ) ⊂ M(bd P ) in the topology of weak convergence is contained in W(P ), i.e., if (µ i ) i∈N is a sequence of polyhedral perturbations of P and µ i w → µ, then µ ∈ W(P ).

1
be a chain of faces that contains an element of every intermediate dimension d ∈ {dim F, . . ., n − 1}.Let i ∈ N.An iterated application of Lemma 5.4, using the triangle inequality for ∥ • ∥ W , and, finally, an application of Lemma 5.3 yields a polyhedral convex function g
Corollary 7.3.Let P ∈ P n n be an isotropic local maximizer of the isotropic constant and let π W be as in Proposition 7.2.Then π W (x → ∥x∥ 2 2 − n − 2) = 0 ∈ L 2 (bd P, vol Φ ).Let F ⊂ P be a facet and X be uniformly distributed on F .Using that (7.1) holds with equality for the signed measures µ i ∈ M(F ) with densities x → x i , i ∈ [n], we have E[∥X∥ 2 2 X] = (n + 2)E[X].