On the equivalence between two problems of asymmetry on convex bodies

Abstract

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 $\max_{K'} |K'|/|K|$? Here, $K'$ 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 [{\rm K. B\"or\"oczky, I. B\'ar\'any, E. Makai Jr. and J. Pach}, Maximal volume enclosed by plates and proof of the chessboard conjecture], Discrete Math. {\bf 69} (1986), 101--120], stating that if the simplex solves (P1) then the simplex solves (P2) as well.