Floating and Illumination Bodies for Polytopes: Duality Results

We consider the question how well a floating body can be approximated by the polar of the illumination body of the polar. We establish precise convergence results in the case of centrally symmetric polytopes. This leads to a new affine invariant which is related to the cone measure of the polytope.


Introduction
Floating bodies are a fundamental notion in convex geometry. Early notions of floating bodies are motivated by the physical description of floating objects. The first systematic study of floating bodies appeared 1822 in a work by C. Dupin [7] on naval engineering. Blaschke, in dimensions 2 and 3, and later Leichtweiss in higher dimensions, used the floating body in the study of affine differential geometry, in particular affine surface area (see [3], [10]). Affine surface area is among the most powerful tools in convexity. It is widely used, for instance in approximation of convex bodies by polytopes, e.g., [4,16,21] and affine differential geometry, e.g., [1,9,22,23].
Dupin's floating body may not exist and related to that, originally affine surface area was only defined for sufficiently smooth bodies. Schütt and Werner [20] and independently Bárány and Larman [2] introduced the convex floating body which, in contrast to Dupin's original definition, always exists and coincides with Dupin's floating body if the latter exists. This, in turn, allowed to define affine surface area for all convex bodies as carried out in [20].
The illumination body was introduced much later in [24] as a further tool to study affine properties of convex bodies. It was pointed out in [26] that the convex floating body and the illumination body are dual notions in the sense that the polar of the floating body and the illumination body of the polar should be close both, in a conceptual and geometric sense. In [13] the duality relation is studied in detail for C 2 + -bodies, i.e. bodies with twice differentiable boundary and everywhere positive curvature, and n p -balls. The paper provides asymptotic sharp estimates how well the polar of the floating body can be approximated by an illumination body of the polar. A limiting procedure leads to a new affine invariant that is different from the affine surface area. It is related to the cone measure of the convex body. These measures play a central role in many aspects of convex geometry, e.g., [5,6,14,15].
The purpose of this paper is to make the duality relation between floating body and illumination body precise when the convex body is a polytope P. It was shown by Schütt [19] that the limit of the (appropriately normalized) volume difference of a polytope P and its floating body leads to a quantity related to the combinatorial structure of the polytope, namely the flags of P (see section 5). Likewise, as shown in [25], the limit of the (appropriately normalized) volume difference of a simplex and its illumination body is related to the combinatorics of the boundary. Now, as in the smooth case [13], a limit procedure leads to a new affine invariant that is not related to the combinatorial structure of the boundary of the polytope, but, as in the smooth case, to cone measures.
The techniques in [13] rely on comparing "extremal" directions, i.e., directions where the boundary of the convex body and its floating body, and the illumination body of its polar, differ the most and the least. The techniques used in [13] employ tools from differential geometry which is possible due to the C 2 + smoothness assumptions. Such tools are no longer available in our present setting of polytopes and we have to use a completely different approach.
It would be interesting to have results for more general classes of convex bodies other than polytopes and the ones investigated in [13]. A major obstruction is that in general it is hard to compute the polar body and even harder to compute the floating body if we do not have smoothness assumptions or are in the case of polytopes. The smooth case [13] and the polytope case seem to be the extremal cases. Indeed, ellipsoids are C 2 + -bodies where equality of the polar of the floating body and the illumination body of the polar can be achieved. And we show here that polytopes display the worst behavior one can expect. Furthermore, the limit is not continuous with respect to the polytope involved since it depends only on the local structure of the boundary (see section 6).
The paper is organized as follows. In the next section we present the main theorem and some consequences. In Section 3 we give the necessary background material and several lemmas needed for the proof of the main theorem. In section 5 we discuss properties of the new affine invariant. We show, with an example, that it is not continuous with respect to the Hausdorff distance. We also show that for this invariant the combinatorial structure of the polytope is less relevant. The relation to the cone measures is the dominant feature. In the final section we address questions of approximation of the floating body by the polar of the illumination of the polar. We show that our convergence results are of pointwise nature and we derive a uniform upper bound for general convex bodies.

Main theorem and consequences
We work in a similar framework as in [13]. We first recall the notions and definitions that we will need. Let K be a n-dimensional convex body and δ ≥ 0. The convex floating body K δ of K was introduced by Schütt and Werner [20] and independently by Barany and Larman [2] as The illumination body K δ of K was defined by E. Werner [24] as If 0 is in the interior of K, the polar K • of K is given by It is a simple fact of convex geometry that for a hyperplane H with corresponding halfspaces H + , H − there is a corresponding point x H ∈ R n such that whenever 0 is in the interior of K ∩ H + . As noted in [26], this polarity relation gives rise to the idea that cutting off with hyperplanes sets of a certain volume of a convex body and including points such that their convex hull with the convex body has a certain volume, should be dual operations, In the same paper it is pointed out that equality cannot be achieved in general since the floating body is always strictly convex and the illumination body of a polytope is always a polytope. As in [13] we like to measure how "close" these two bodies are in the polytope case. A further outcome of such a study shows how well the floating body of a polytope can be approximated by a polytope, namely the polar of an illumination of the polar. For x ∈ R n \{0} and K a convex body with 0 ∈ K we denote by r K (x) = sup{λ ≥ 0 : λ x ∈ K} the radial function of K. To measure how close two centrally symmetric convex bodies S 1 , S 2 are, we use the distance It is worthwhile to mention that log d(·, ·) is a metric on the space of convex bodies in R n which induces the same topology as the Hausdorff distance.
For a centrally symmetric convex body S and 0 < δ < 1 2 , we put S δ = (S • ) δ • . We then define Please note that d L(S) (δ ) = d S (δ ) for every linear invertible map L. Observe also that d B n 2 (δ ) = 1 . One of the main theorems in [13] states that for origin symmetric convex bodies C in R n that are C 2 + , i.e. the Gauss curvature κ(x) exists for every x ∈ ∂C and is strictly positive, the relation (2.4) can be made precise in terms of the cone measures of C and C • .
For a Borel set A ⊂ ∂C, the cone measure M C of A is defined as M C (A) = |conv[0, A]| n . The density function of M C is m C (x) = 1 n x, N(x) (see [15]), and we write n C (x) = 1 n|C| n x, N(x) for the density of the normalized cone measure P C of C (again see e.g., [15]). This means that, e.g., [15], Denote by N C : ∂C → S n−1 , x → N(x) the Gauss map of C, see e.g., [18]. Then, similarly, x,N(x) n is the density function of the "cone measure" x,N(x) n is the density of the normalized cone measure P C • of C • , see, e.g., [15]. When C is C 2 + , this formula holds for all x ∈ ∂C. Thus As observed in [13], we then have for a centrally symmetric C 2 + convex body C that In the case of a polytope the (discrete) densities n P and n P • of the normalized cone measures can be expressed as follows. Let ξ be an extreme point of P. Let F ξ be the facet of P • that has ξ as an outer normal. The (discrete) density of the normalized cone measure of P • is where · denotes the standard Euclidean norm on R n . Let s(F ξ ) be the (n − 1)-dimensional Santaló point (see, e.g., [8,18] Let C ξ be the cone with base F ξ and apex at the origin and let The expression n P (ξ ) is the ratio of this volume and the volume of P. We see n P (ξ ) as a cone measure associated to ξ since this volume is the cone measure of the set of all points of Z with outer normal ξ ξ .
Our main theorem can be expressed in terms of n P (ξ ) and n P • (ξ ) and reads as follows.
Theorem 2.1 Let P ⊆ R n be a centrally symmetric polytope. Then Recall that for 1 ≤ p < ∞, the n p -unit balls are defined as B n , is an immediate consequence of Theorem 2.1.

Tools and Lemmas
Let P ⊆ R n be a centrally symmetric polytope. In [11] it was shown that for centrally symmetric convex bodies Dupin's floating body exists and coincides with the convex floating body. This means that every support hyperplane of P δ cuts off the volume δ |P| n from P. We use this fact throughout the remainder of the paper. We denote by ext(P) the set of extreme points of P. Note that this set coincides with the set of vertices of P. For ξ ∈ ext(P), let F 1 , . . . , F k be the (n − 1)-dimensional facets of P such that ξ ∈ F i . Then there are y 1 , . . . , y k ∈ R n such that for 1 ≤ i ≤ k, Observe that y 1 , . . . , y k are vertices of P • and that F ξ := conv[y 1 , . . . , For δ > 0, let P δ be the floating body of P. Let ξ ∈ ext(P). We denote by ξ δ the unique point in the intersection of ∂ P δ with the line segment [0, ξ ] and by x δ the unique point in the intersection of ∂ P δ with [0, ξ ]. We denote by ξ δ the unique point such that ξ is the unique point in the intersection of ∂ P δ with [0, ξ δ ].
The next lemma provides a formula for ξ δ ξ if δ > 0 is sufficiently small.
Lemma 3.1 Let P ⊆ R n be a centrally symmetric polytope. Then there is δ 0 > 0 such that for every 0 ≤ δ ≤ δ 0 and every vertex ξ ∈ ∂ P we have Proof. Let e i ∈ R n be the vector with i-th entry 1 and the other entries are 0. We first consider the case that ξ = e n and s( is an (n − 1)-dimensional convex body with centroid in the origin. For self similarity reasons the where the polar is taken in R n−1 . Then s(F) = 0 and It is a well-known fact (see [17]) that for a convex body C we have the identities It follows immediately that the centroid of lies in the origin. The volume of the cone with base B × {0} and apex ξ = e n is given by |B| n−1 /n. Let 0 ≤ ∆ ≤ 1. Then the volume of the cone with base and apex e n is given by ∆ n n |B| n−1 . There is ∆ 0 > 0 such that for every 0 ≤ ∆ ≤ ∆ 0 the point e n is the only vertex of P contained in the half-space Hence, the above described cone is given by or, equivalently, Choose δ 0 > 0 sufficiently small such that for every 0 ≤ δ ≤ δ 0 the value of ∆ is smaller than or equal to ∆ 0 . It was shown in [11] that for centrally symmetric convex bodies, the floating body coincides with the convex floating body. Thus, since P is centrally symmetric, the floating body of P coincides with the convex floating body, and therefore the hyperplane {x ∈ R n : x n ≥ 1 − ∆} touches P δ at the centroid of This centroid is given by For a general vertex ξ and general s(F ξ ) note first that ξ , s(F ξ ) = 1 and thus, ξ ∈ s(F ξ ) ⊥ . Let L ∈ R n×n be a matrix with last row s(F x i ) and the other rows are a basis of ξ ⊥ . Let L −t the inverse of the transpose. Since ξ , s(F ξ ) = 1 it follows that L is a full rank matrix with L(ξ ) = e n and L −t (s(F ξ )) = e n . Then, LP is a centrally symmetric polytope with vertex L(ξ ) = e n and s(F L(ξ ) ) = s(F e n ) = e n . Note that ξ δ ξ = (L(ξ )) δ L(ξ ) . The lemma follows from The second equality follows from the fact that for every (n − 1)-dimensional vector space V with normal u, every linear invertible map S : R n → R n and every Borel set For a vertex ξ ∈ P, ξ δ is the unique point in the intersection of ∂ P δ and the line segment [0, ξ ].
Lemma 3.2 Let P be a centrally symmetric polytope. Then there is a δ 0 > 0 such that for every 0 ≤ δ ≤ δ 0 and every extreme point ξ ∈ ext(P) we have Proof. We show that is a support hyperplane of (P • ) δ . The lemma then follows immediately. Let z ∈ F ξ and ∆ ≥ 0. The volume of the cone with base F ξ and apex z + ∆ ξ ξ is 1 n |F ξ | n−1 ∆. There is a ∆ 0 > 0 and an η > 0 such that has non-empty relative interior and such that for every z ∈ F η ξ and every 0 ≤ ∆ ≤ ∆ 0 we have Let δ 0 > 0 be such that n|P • | n |F ξ | n−1 δ ≤ ∆ 0 for every 0 ≤ δ ≤ δ 0 . It is obvious that for every z ∈ F η ξ the vector lies on the boundary of (P • ) δ . Since F ξ is contained in the hyperplane {y ∈ R n : y, ξ = 1}, it follows that is a support hyperplane of (P • ) δ . Lemma 3.3 Let P ⊆ R n be a centrally symmetric polytope. Then there is a δ 0 > 0 such that for every Proof. The first inclusion is obvious. Choose δ 0 > 0 as in Lemma 3.2.
Consider the set

Proof of Theorem 2.1 and Corollary 2.2
We recall the quantities that are relevant for our main theorem. For ξ ∈ ext(P), we put  We split the proof of the theorem and show separately the upper and lower bound.

Upper bound
We prove the following proposition. Proof. Let c 0 ≥ 0 be such that G(P) = G c 0 (P) and put δ = c 0 δ 1/n . By Lemma 3.2, Lemma 3.4 and Lemma 3.5, a sufficient condition for P δ ⊆ a P δ is that for every ξ ∈ ext(P). Hence, By Lemma 3.1, Lemma 3.2 and Corollary 3.4, a sufficient condition for P δ ⊆ aP δ is that for every ξ ∈ ext(P) and that 1 2 From the first condition we derive that for every ξ ∈ ext(P). By Lemma 3.6 there is a constant k > 0 and δ 0 > 0 such that for every 0 ≤ δ ≤ δ 0 we have and we may assume that k and δ 0 are taken uniformly with respect to all pairs (ξ , ξ ). Hence, for δ ≤ δ 0 we have the condition that a ≥ 1 We check that all three conditions are met if one takes a = 1 + G(P) δ 1 n (1 + o(1)). The condition a ≥ 1 ≤1 + G c 0 (P)δ 1/n + o(δ 1/n ) ≤ 1 + G(P)δ 1/n + o(δ 1/n ). Finally, the condition

Lower Bound
We prove the following proposition.

Proof of Corollary 2.2
We first treat the case of the cube. It is well known that the volume product |S n−1 | n−1 |(S n−1 ) • | n−1 of the (n − 1)-dimensional simplex is n n ((n−1)!) 2 . Hence, as F ξ is an (n − 1)-dimensional regular simplex, Therefore, The minimum over all which completes the proof.
Now we show the statement of Corollary 2.2 in the case of the crosspolytope.

The combinatorial structure of d P
In [19], it was proved that the following relation holds for all polytopes P ⊆ R n , where fl n (P) denotes the number of flags of P. A flag of P is an (n + 1)-tuple (F 0 , . . . , F n ) such that F i is an i-dimensional face of P and F 0 ⊂ F 1 ⊂ · · · ⊂ F n .
This theorem suggests that also d P , and hence G(P), might only depend on the combinatorial structure of P. The fact that d P is invariant under affine transformations of P supports this conjecture. However, this is not the case, as is illustrated by the following 2-dimensional example.

Approximation results for the floating body and open questions
The parameter d S measures the best approximation of the floating body by the polar of an illumination body of the polar. We establish a uniform bound for this quantity, independent of the convex body.
Proposition 6.1 Let S ⊆ R n be a centrally symmetric convex body. Then there exist constants G n , δ n only depending on the dimension such that In particular, the proposition yields Thus, δ 1/n is already the worst order of convergence we can get in general. The polytopal case shows that we cannot hope for any better uniform rate of convergence. Proposition 6.1 does not involve the floating body any more. We address the question if we get a better uniform bound if we involve the illumination body. At best, how does the optimal G n look like such that d S (δ ) ≤ 1 + G n δ 1/n + o(δ 1/n ) where the error term o(δ 1/n ) does only depend on the dimension? It would be interesting to know about the maximizers if they exist. Furthermore, it would be interesting to know something about the best uniform bound on subclasses like polytopes, C 1 -bodies or C 2 -bodies.
Proof of Proposition 6.1. The quantity d(S δ , S) is invariant with respect to linear transformations and we may therefore assume that S is in John position, i.e., in particular, Let ξ ∈ ∂ S. Then 1 ≤ ξ ≤ √ n. Put B Taking into account that |S| n ≤ √ n n |B n 2 | and ξ ≥ 1 we obtain ξ δ ξ ≥ 1 − √ n n|B n 2 | n |B n−1 2 | n−1 The desired result follows.
One might ask if the convergence result in Theorem 2.1 is uniform, i.e., does |d P (δ ) − 1 − G(P)δ 1/n | ≤ o(δ 1/n ) hold with an error term o(δ 1/n ) only depending on the dimension? This is not the case. Indeed, the floating and illumination bodies are stable with respect to the Hausdorff distance, [12] and hence, with respect to the distance d. This means that if We also like to address the problem to compute the optimal constantG(P) such that d P (δ ) ≤ 1 +G(P)δ 1/n + o(δ 1/n ) for centrally symmetric polytopes such that o(δ 1/n ) is only a dimension dependent error. The problem of proving such a result is already illustrated by the example P(ε) of Section 5. The facet and vertex structure of a polytope is not stable with respect to the distance d but the convergence result Theorem 2.1 depends highly on these quantities. On the other hand P(ε) is close to B n 1 for small ε and therefore, d P(ε) (δ ) and d B n 1 (δ ) behave similarly for a wide range of δ not to close to 0. Deriving a uniform bound would demand techniques which take into account the global structure of the convex bodies.