Mathieu Meyer

Sections of the unit ball of Ipn

Let Bnp={(xi)∈RN; σi=li=n|xi|p⩽1}, 1+⩽p⩽+∞, and let Ek be a k-dimensional subspace of Rn; it is proved that if p ⩾ 2 (resp. ⩽) then vol(Bpn ∩ Ek) ⩾ vol(Bpk) (resp. ⩽). We give some applications to linear forms.

ZONOIDS WITH MINIMAL VOLUME-PRODUCT- A NEW PROOF

A new and simple proof of the following result is given: The product of the volumes of a symmetric zonoid A in RI and of its polar body is minimal if and only if A is the Minkowski sum of n segments.

The Santaló-regions of a convex body

Motivated by the Blaschke-Santalo inequality, we define for a convex body K in R and for t ∈ R the Santalo-regions S(K,t) of K. We investigate properties of these sets and relate them to a concept of Affine Differential Geometry, the affine surface area of K. Let K be a convex body in R. For x ∈ int(K), the interior of K, let K be the polar body of K with respect to x. It is well known that there exists a unique x0 ∈ int(K) such that the product of the volumes |K||K0 | is minimal (see for instance [Sch]). This unique x0 is called the Santalo-point of K. Moreover the Blaschke-Santalo inequality says that |K||K0 | ≤ v n (where vn denotes the volume of the n-dimensional Euclidean unit ball B(0, 1)) with equality if and only if K is an ellipsoid. For t ∈ R we consider here the sets S(K, t) = {x ∈ K : |K||K | v2 n ≤ t}. Following E. Lutwak, we call S(K, t) a Santalo-region of K. Observe that it follows from the Blaschke-Santalo inequality that the Santalopoint x0 ∈ S(K, 1) and that S(K, 1) = {x0} if and only if K is an ellipsoid. Thus S(K, t) has non-empty interior for some t < 1 if and only if K is not an ellipsoid. In the first part of this paper we show some properties of S(K, t) and give estimates on the “size” of S(K, t). This question was asked by E. Lutwak. ∗the paper was written while both authors stayed at MSRI †supported by a grant from the National Science Foundation. MSC classification 52

Une Caracterisation Volumique de Certains Espaces Normes de Dimension Finie

We give a geometric characterization of finite dimensional normed spacesE, with a 1-unconditional basis, such that their volumetric product is minimal.

Shadow Systems and Volumes of Polar Convex Bodies

It is proved that the reciprocal of the volume of the polar bodies, about the Santalo point, of a shadow system of convex bodies K t , is a convex function of t , thus extending to the non-symmetric case a result of Campi and Gronchi. The case that the reciprocal of the volume is an affine function of t is also investigated and is characterized under certain conditions. These results are applied to prove an exact reverse Santalo inequality for polytopes in ℝ d that have at most d + 3 vertices.

Convex bodies with minimal volume product in ℝ2

We give a new proof of the following result due to Mahler: the product of the surface areas of a convex body in the plane and of its polar body is always bigger than that of a triangle; and we establish the case of equality.

Constructing a Polytope to Approximate a Convex Body

We develop an algorithm to construct a convex polytopeP withn vertices, contained in an arbitrary convex bodyK inRd, so that the ratio of the volumes |K/P|/|K| is dominated byc ·. d/n2/(d−1).

Some inequalities about mixed volumes

AbstractWe prove inequalities about the quermassintegralsVk(K) of a convex bodyK in ℝn (here,Vk(K) is the mixed volumeV((K, k), (Bn,n − k)) whereBn is the Euclidean unit ball). (i) The inequality

The volume of the intersection of a convex body with its translates

\frac{{V_k \left( {K + L} \right)}}{{V_{k - 1} \left( {K + L} \right)}} \geqslant \frac{{V_k \left( K \right)}}{{V_{k - 1} \left( K \right)}} + \frac{{V_k \left( L \right)}}{{V_{k - 1} \left( L \right)}}