This is an announcement for the paper "On the equivalence between two problems of asymmetry on convex bodies" by Christos Saroglou.
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"{o}r"{o}czky, I. B'{a}r'{a}ny, 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.
Archive classification: math.FA
Remarks: Submitted for publication, November 2013
Submitted from: saroglou@math.tamu.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1311.4955
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