Concentration of the Intrinsic Volumes of a Convex Body

TL;DR

This paper investigates how the intrinsic volumes of convex bodies concentrate around a central value and shows that scaled cubes maximize entropy in this sequence, using probabilistic and information-theoretic methods.

Contribution

It introduces a probabilistic and information-theoretic framework to analyze the concentration and entropy of intrinsic volume sequences of convex bodies.

Findings

Intrinsic volume sequences sharply concentrate around a central index.

Scaled cubes maximize entropy among convex bodies with fixed central intrinsic volume.

The study provides new insights into the geometric and probabilistic structure of convex bodies.

Abstract

The intrinsic volumes are measures of the content of a convex body. This paper uses probabilistic and information-theoretic methods to study the sequence of intrinsic volumes of a convex body. The main result states that the intrinsic volume sequence concentrates sharply around a specific index, called the central intrinsic volume. Furthermore, among all convex bodies whose central intrinsic volume is fixed, an appropriately scaled cube has the intrinsic volume sequence with maximum entropy.