An improved stability result for Gr\"unbaum's inequality

TL;DR

This paper improves the stability estimate for Gr"unbaum's inequality, showing that bodies nearly achieving the inequality are closer to cones than previously established, with a better exponent in the stability bound.

Contribution

The paper enhances Groemer's stability estimate for Gr"unbaum's inequality by increasing the exponent from 1/(2n^2) to 1/(2n), providing a sharper measure of how close bodies are to cones.

Findings

Improved the stability exponent from 1/(2n^2) to 1/(2n).

Bodies nearly attaining Gr"unbaum's inequality are closer to cones than previously known.

The result refines the quantitative understanding of the equality case in Gr"unbaum's inequality.

Abstract

Given a hyperplane $H$ cutting a compact, convex body $K$ of positive Lebesgue measure through its centroid, Gr\"unbaum proved that $$\frac{|K\cap H^+|}{|K|}\geq \left(\frac{n}{n+1}\right)^n,$$ where $H^+$ is a half-space of boundary $H$. The inequality is sharp and equality is reached only if $K$ is a cone. Moreover, bodies that almost achieve equality are geometrically close to being cones, as Groemer showed in 2000 by giving his stability estimates for Gr\"unbaum's inequality. In this paper, we improve the exponent in the stability inequality from Groemer's $\frac{1}{2n^2}$ to $\frac{1}{2n}$.