Some new minimax inequalities for centered convex bodies

The purpose of this manuscript is to derive some new minimax inequalities for centered convex bodies in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}<^>{d}$$\end{document}. As consequences, new characterizations of ellipsoids will be established as well. Some open problems related to the Busemann-Petty list of problems will also be discussed.