A generalization of Gr\"unbaum's inequality in RCD$(0,N)$-spaces

TL;DR

This paper extends Gr"unbaum's inequality from convex geometry to curved spaces with nonnegative Ricci curvature, providing new volume bounds, rigidity results, and stability analysis in the setting of RCD spaces and weighted manifolds.

Contribution

It generalizes Gr"unbaum's inequality to RCD(0,N)-spaces and weighted manifolds, introducing a volume bound involving Busemann functions and establishing rigidity and stability results.

Findings

Volume bounds for convex sets in RCD spaces

Rigidity results via localization method

Stability analysis of the inequality

Abstract

We generalize Gr\"unbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to $\mathrm{RCD}(0,N)$-spaces with $N \in (1,\infty)$ as well as weighted Riemannian manifolds of $\mathrm{Ric}_N \ge 0$ for $N \in (-\infty,-1) \cup \{\infty\}$. Our formulation makes use of the isometric splitting theorem; given a convex set $\Omega$ and the Busemann function associated with any straight line, the volume of the intersection of $\Omega$ and any sublevel set of the Busemann function that contains a barycenter of $\Omega$ is bounded from below in terms of $N$. We also extend this inequality beyond uniform distributions on convex sets. Moreover, we establish some rigidity results by using the localization method, and the stability problem is also studied.