Sharp inequalities for the mean distance of random points in convex bodies

TL;DR

This paper establishes optimal bounds for the ratio of the mean distance between two random points in a convex body to its mean width, using advanced inequalities and optimization techniques.

Contribution

It derives sharp bounds for the ratio of mean distance to mean width in convex bodies, extending and connecting existing geometric inequalities.

Findings

Derived optimal bounds for the ratio of mean distance to mean width.

Identified extremal cases where bounds are attained.

Connected results with classical inequalities and geometric measures.

Abstract

For a convex body $K\subset\mathbb{R}^d$ the mean distance $\Delta(K)=\mathbb{E}|X_1-X_2|$ is the expected Euclidean distance of two independent and uniformly distributed random points $X_1,X_2\in K$. Optimal lower and upper bounds for ratio between $\Delta(K)$ and the first intrinsic volume $V_1(K)$ of $K$ (normalized mean width) are derived and degenerate extremal cases are discussed. The argument relies on Riesz's rearrangement inequality and the solution of an optimization problem for powers of concave functions. The relation with results known from the existing literature is reviewed in detail.