A generalization of Grünbaum's inequality in RCD(0,N)-spaces

We generalize Grünbaum'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 Ω and the Busemann function associated with any straight line, the volume of the intersection of Ω and any sublevel set of the Busemann function that contains a barycenter of Ω 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.