A general functional version of Grünbaum's inequality

Abstract

A classical inequality by Grünbaum provides a sharp lower bound for the ratio $\mathrm{vol}\left(K^{-}\right)/\mathrm{vol}\left(K\right)$, where $K^{-}$ denotes the intersection of a convex body with non-empty interior $K\subset R^n$ with a halfspace bounded by a hyperplane H passing through the centroid $g\left(K\right)$ of K.

In this paper we extend this result to the case in which the hyperplane H passes by any of the points lying in a whole uniparametric family of r-powered centroids associated to K (depending on a real parameter $r\ge 0$), by proving a more general functional result on concave functions.

The latter result further connects (and allows one to recover) various inequalities involving the centroid, such as a classical inequality (due to Minkowski and Radon) that relates the distance of $g\left(K\right)$ to a supporting hyperplane of K, or a result for volume sections of convex bodies proven independently by Makai Jr. & Martini and Fradelizi.