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arXiv · 1804.05009

On the reverse isodiametric problem and Dvoretzky-Rogers-type volume bounds

Bernardo González Merino·Matthias Schymura
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Abstract

The isodiametric inequality states that the Euclidean ball maximizes the volume among all convex bodies of a given diameter. We are motivated by a conjecture of Makai Jr.~on the reverse question: Every convex body has a linear image whose isodiametric quotient is at least as large as that of a regular simplex. We relate this reverse isodiametric problem to minimal volume enclosing ellipsoids and to the Dvoretzky-Rogers-type problem of finding large volume simplices in any decomposition of the identity matrix. As a result, we solve the reverse isodiametric problem for $o$-symmetric convex bodies and obtain a strong asymptotic bound in the general case. Using the Cauchy-Binet formula for minors of a product of matrices, we obtain Dvoretzky-Rogers-type volume bounds which are of independent interest.

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Bernardo González Merino, Matthias Schymura. 2018-04-13. On the reverse isodiametric problem and Dvoretzky-Rogers-type volume bounds. https://arxiv.org/abs/1804.05009

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