Random section and random simplex inequality

TL;DR

This paper introduces a generalized inequality relating the expected volume of random sections of convex bodies to the convex hulls of random points, unifying and extending several classical geometric inequalities.

Contribution

It establishes a new inequality connecting random sections and convex hulls, with conditions for equality, generalizing multiple known inequalities in convex geometry.

Findings

Equality holds iff $K$ is an origin-centered ellipsoid for $k>1$

Reduces to Busemann intersection and simplex inequalities for specific parameters

Provides an affine version generalizing Schneider and Blaschke-Gr"omer inequalities

Abstract

Consider some convex body $K\subset\mathbb R^d$. Let $X_1,\dots, X_k$, where $k\leq d$, be random points independently and uniformly chosen in $K$, and let $\xi_k$ be a uniformly distributed random linear $k$-plane. We show that for $p\geq-d+k+1$, \[ \mathbb E\,|K\cap\xi_k|^{d+p}\leq c_{d,k,p} \cdot|K|^k\, \,\mathbb E\,|\mathrm{conv}(0,X_1, \dots,X_k)|^p, \] where $|\cdot|$ and $\mathrm{conv}$ denote the volume of correspondent dimension and the convex hull. The constant $c_{d,k,p}$ is such that for $k>1$ the equality holds if and only if $K$ is an ellipsoid centered at the origin, and for $k=1$ the inequality turns to equality. If $p=0$, then the inequality reduces to the Busemann intersection inequality, and if $k=d$ -- to the Busemann random simplex inequality. We also present an affine version of this inequality which similarly generalizes the Schneider inequality and the Blaschke-Gr\"omer inequality.