A general functional version of Grünbaum's inequality

Alonso-Gutiérrez, David (Universidad de Zaragoza) ; Marín Sola, Francisco ; Martín Goñi, Javier ; Yepes Nicolás, Jesús

Resumen: A 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.

Idioma: Inglés

DOI: 10.1016/j.jmaa.2024.129065

Año: 2025

Publicado en: JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 544, 1 (2025), 129065 [20 pp.]
ISSN: 0022-247X

Tipo y forma: Article (Published version)

Área (Departamento): Área Análisis Matemático (Dpto. Matemáticas)