000146910 001__ 146910 000146910 005__ 20241129141114.0 000146910 0247_ $$2doi$$a10.1016/j.jmaa.2024.129065 000146910 0248_ $$2sideral$$a140718 000146910 037__ $$aART-2025-140718 000146910 041__ $$aeng 000146910 100__ $$0(orcid)0000-0003-1256-3671$$aAlonso-Gutiérrez, David$$uUniversidad de Zaragoza 000146910 245__ $$aA general functional version of Grünbaum's inequality 000146910 260__ $$c2025 000146910 5060_ $$aAccess copy available to the general public$$fUnrestricted 000146910 5203_ $$aA 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. 000146910 540__ $$9info:eu-repo/semantics/openAccess$$aby-nc$$uhttp://creativecommons.org/licenses/by-nc/3.0/es/ 000146910 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/publishedVersion 000146910 700__ $$aMarín Sola, Francisco 000146910 700__ $$aMartín Goñi, Javier 000146910 700__ $$aYepes Nicolás, Jesús 000146910 7102_ $$12006$$2015$$aUniversidad de Zaragoza$$bDpto. Matemáticas$$cÁrea Análisis Matemático 000146910 773__ $$g544, 1 (2025), 129065 [20 pp.]$$pJ. math. anal. appl.$$tJOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS$$x0022-247X 000146910 8564_ $$s399533$$uhttps://zaguan.unizar.es/record/146910/files/texto_completo.pdf$$yVersión publicada 000146910 8564_ $$s1754509$$uhttps://zaguan.unizar.es/record/146910/files/texto_completo.jpg?subformat=icon$$xicon$$yVersión publicada 000146910 909CO $$ooai:zaguan.unizar.es:146910$$particulos$$pdriver 000146910 951__ $$a2024-11-29-13:24:12 000146910 980__ $$aARTICLE