On the equivalence between two problems of asymmetry on convex bodies

2013/11/20 by Christos Saroglou, Saroglou, Christos
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA

paper · pdf · doi:10.48550/arxiv.1311.4955

A serious logical gap fixed, from the previous version

arxiv created 2014/11/22 · arxiv updated 2014/11/25

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. Böröczky, I. Bárány, 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.

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