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On the Equivalence Between Two Problems of Asymmetry on Convex Bodies

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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 Böröczky et al. (Discrete Math 69:101–120, 1986), stating that if the simplex solves (P1), then the simplex solves (P2) as well.

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Acknowledgments

I would like to thank the referee(s) for many helpful suggestions and improvements, especially for discovering a serious logical gap in a previous version of this manuscript, particularly in the following statement: every solution for Problem 1.1 is a limit of solutions for Problem 1.1, restricted in the class of polytopes with at most N facets, as \(N\rightarrow \infty \).

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  1. Department of Mathematics, Texas A&M University, College Station, TX, 77840, USA

    Christos Saroglou

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Saroglou, C. On the Equivalence Between Two Problems of Asymmetry on Convex Bodies. Discrete Comput Geom 54, 573–585 (2015). https://doi.org/10.1007/s00454-015-9722-x

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