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

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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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Acknowledgements

We thank Ambros Gleixner, Benjamin Müller, and Felipe Serrano from the Zuse Institute Berlin for discussions and help regarding the scip experiments described in Remark 4.6. Furthermore, we thank Peter Gritzmann and Martin Henk for the possibility of mutual research visits at TU Munich and TU Berlin. The first author also thanks TU Munich and the University Centre of Defence of San Javier, where part of this work has been done. We would like to thank the anonymous referee for comments and suggestions that helped us to improve the writing, and in particular, for pointing out that [6, Cor. 3] is incorrectly stated.

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Authors and Affiliations

  1. Departamento de Didáctica de las Ciencias Matemáticas y Sociales, Facultad de Educación, Universidad de Murcia, 30100, Murcia, Spain

    Bernardo González Merino

  2. BTU Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046, Cottbus, Germany

    Matthias Schymura

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  1. Bernardo González Merino
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  2. Matthias Schymura
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Correspondence to Bernardo González Merino.

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The research of the Bernardo González Merino is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. He was partially supported by Fundación Séneca project 19901/GERM/15, MINECO project MTM2015-63699-P, and MICINN Project PGC2018-094215-B-I00, Spain. The second author was partially supported by the Freie Universität Berlin within the Excellence Initiative of the German Research Foundation.

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Merino, B.G., Schymura, M. On the reverse isodiametric problem and Dvoretzky–Rogers-type volume bounds. RACSAM 114, 136 (2020). https://doi.org/10.1007/s13398-020-00867-7

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