A generalization of Gr\"unbaum's inequality

TL;DR

This paper generalizes Gr"unbaum's inequality to hyperplanes not passing through the centroid, providing new bounds and an application to comparing convex body sections with their maximal parallel sections.

Contribution

It extends Gr"unbaum's inequality to a broader class of hyperplanes and derives a sharp inequality relating convex body sections to their maximal parallel sections.

Findings

Generalized Gr"unbaum's inequality for arbitrary hyperplanes

Derived a sharp inequality for convex body sections

Established bounds comparing sections to maximal parallel sections

Abstract

Gr\"unbaum's inequality gives sharp bounds between the volume of a convex body and its part cut off by a hyperplane through the centroid of the body. We provide a generalization of this inequality for hyperplanes that do not necessarily contain the centroid. As an application, we obtain a sharp inequality that compares sections of a convex body to the maximal section parallel to it.