Applications of Gr\"unbaum-type inequalities

TL;DR

This paper establishes exact inequalities relating intrinsic and dual volumes of convex bodies intersected with subspaces, extending Gr"unbaum-type inequalities and generalizing previous results for sections and projections.

Contribution

It introduces new inequalities for intrinsic and dual volumes of convex bodies, extending Gr"unbaum's inequalities to broader settings and higher dimensions.

Findings

Proved inequalities for intrinsic volumes of convex body intersections with subspaces.

Extended Gr"unbaum's inequality to dual volumes and higher dimensions.

Generalized inequalities for sections and projections of convex bodies.

Abstract

Let $1\leq i \leq k < n$ be integers. We prove the following exact inequalities for any convex body $K\subset\mathbb{R}^n$ with centroid at the origin, and any $k$-dimensional subspace $E\subset \mathbb{R}^n$: \begin{align*} &V_i \big( K\cap E \big) \geq \left( \frac{i+1}{n+1} \right)^i \max_{x\in K} V_i \big( ( K-x) \cap E \big) , \\ &\widetilde{V}_i \big( K\cap E \big) \geq \left( \frac{i+1}{n+1} \right)^i \max_{x\in K} \widetilde{V}_i \big( ( K-x) \cap E \big) ; \end{align*} $V_i$ is the $i$th intrinsic volume, and $\widetilde{V}_i$ is the $i$th dual volume taken within $E$. Our results are an extension of an inequality of M. Fradelizi, which corresponds to the case $i=k$. Using the same techniques, we also establish extensions of "Gr\"unbaum's inequality for sections" and "Gr\"unbaum's inequality for projections" to dual volumes.