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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 r0), 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.
Alonso–Gutiérrez et al. (Fri,) studied this question.