Minimal Volume Product of Three Dimensional Convex Bodies with Various Discrete Symmetries

The volume product of convex bodies with many hyperplane symmetries

Mahler’s conjecture predicts a sharp lower bound on the volume of the polar of a convex body in terms of its volume. We confirm the conjecture for convex bodies with many hyperplane symmetries in the following sense: their hyperplanes of symmetries have a one-point intersection. Moreover, we obtain improved sharp lower bounds for classes of convex bodies which are invariant by certain reflectio...

Convex bodies with minimal volume product in R2 - a new proof

Gaussian Marginals of Convex Bodies with Symmetries

LetK be a convex body in the Euclidean space Rn, n ≥ 2, equipped with its standard inner product 〈·, ·〉 and Euclidean norm | · |, and let μ denote the uniform (normalized Lebesgue) probability measure on K. In this paper we consider k-dimensional marginals of μ, that is, the push-forward μ ◦ P−1 E of μ by the orthogonal projection PE onto some k-dimensional subspace E ⊂ Rn. The question of whet...

Convex Bodies with Minimal Mean Width

is isotropic with respect to an appropriate measure depending on f . The purpose of this note is to provide applications of this point of view in the case of the mean width functional T 7→ w(TK) under various constraints. Recall that the width of K in the direction of u ∈ Sn−1 is defined by w(K,u) = hK(u) + hK(−u), where hK(y) = maxx∈K〈x, y〉 is the support function of K. The width function w(K,...

Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

Given r > 1, we consider the minimal volume, minimal surface area and minimal mean width of convex bodies in E that contain a unit ball, and the extreme points are of distance at least r from the centre of the unit ball; more precisely, we investigate the difference of these minimums and of the volume, surface area and mean width, respectively, of the unit ball. As r tends to one, we describe t...