A general functional version of Gr\"unbaum's inequality

A classical inequality by Gr\"unbaum provides a sharp lower bound for the ratio vol(K⁻)/vol(K), where K⁻ denotes the intersection of a convex body with non-empty interior Kⁿ with a halfspace bounded by a hyperplane H passing through the centroid g(K) 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≥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(K) to a supporting hyperplane of K, or a result for volume sections of convex bodies proven independently by Makai Jr. & Martini and Fradelizi.