The study of high-dimensional convex bodies from a geometric and analytic point of view,
with an emphasis on the dependence of various parameters on the dimension stands at the …

The entropy power inequality, which plays a fundamental role in information theory and
probability, may be seen as an analogue of the Brunn-Minkowski inequality. Motivated by …

Little known relations of the renown concept of the halfspace depth for multivariate data with
notions from convex and affine geometry are discussed. Maximum halfspace depth may be …

A generalized Berwald manifold is a Finsler manifold admitting a linear connection on the
base manifold such that parallel transports preserve the Finslerian length of tangent vectors …

Following ideas of Iriyeh and Shibata we give a short proof of the three-dimensional Mahler
conjecture for symmetric convex bodies. Our contributions include, in particular, simple self …

In the context of his work on maximal functions in the 1980s, Jean Bourgain came across the
following geometric question: Is there c> 0 such that for any dimension n and any convex …

We prove small-deviation estimates for the volume of random convex sets. The focus is on
convex hulls and Minkowski sums of line segments generated by independent random …

We establish dual equivalent forms involving relative entropy, Fisher information, and
optimal transport costs of inverse Santaló inequalities. We show in particular that the Mahler …

Our purpose here is to give an overview of known results and open questions concerning
the volume product 𝒫 (K)= minz∈ K vol (K) vol ((K− z)∗) of a convex body K in ℝn. We …

The affine quermassintegrals associated to a convex body in $\mathbb {R}^ n $ are affine-
invariant analogues of the classical intrinsic volumes from the Brunn–Minkowski theory, and …