doi:10.1016/j.jmaa.2024.129065engAlonso-Gutiérrez, DavidMarín Sola, FranciscoMartín Goñi, JavierYepes Nicolás, JesúsA general functional version of Grünbaum's inequalityART-2025-140718A classical inequality by Grünbaum 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 K ⊂ Rn 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.2025http://zaguan.unizar.es/record/14691010.1016/j.jmaa.2024.129065http://zaguan.unizar.es/record/146910oai:zaguan.unizar.es:146910JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 544, 1 (2025), 129065 [20 pp.]by-nchttp://creativecommons.org/licenses/by-nc/3.0/es/info:eu-repo/semantics/openAccess