(Springer, 2021-03-16) Marín Sola, Francisco; Yepes Nicolás, Jesús; Matemáticas; Facultades de la UMU::Facultad de Matemáticas
Given a compact set K ⊂ R^n of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio vol(K^-)/vol(K), depending on the concavity nature of the function that gives the volumes of crosssections (parallel to H) of K, where K^− denotes the intersection of K with a halfspace bounded by H. When K is convex, this inequality recovers a classical result by Grünbaum. To this respect, we also show that the log-concave case is the limit concavity assumption for such a generalization of Grünbaum’s inequality.
