On the Equivalence Between Two Problems of Asymmetry on Convex Bodies

The simplex was conjectured to be the extremal convex body for the two following “problems of asymmetry”: (P1) What is the minimal possible value of the quantity maxK′|K′|/|K|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\max _{K'} |K'|/|K|$$\end{document}? Here, K′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K'$$\end{document} ranges over all symmetric convex bodies contained in K. (P2) What is the maximal possible volume of the Blaschke body of a convex body of volume 1? Our main result states that (P1) and (P2) admit precisely the same solutions. This complements a result from Böröczky et al. (Discrete Math 69:101–120, 1986), stating that if the simplex solves (P1), then the simplex solves (P2) as well.