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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Isotropic measures and stronger forms of the reverse isoperimetric inequality
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by Károly J. Böröczky and Daniel Hug PDF
Trans. Amer. Math. Soc. 369 (2017), 6987-7019 Request permission

Abstract:

The reverse isoperimetric inequality, due to Keith Ball, states that if $K$ is an $n$-dimensional convex body, then there is an affine image $\tilde {K}$ of $K$ for which $S(\tilde {K})^n/V(\tilde {K})^{n-1}$ is bounded from above by the corresponding expression for a regular $n$-dimensional simplex, where $S$ and $V$ denote the surface area and volume functional. It was shown by Franck Barthe that the upper bound is attained only if $K$ is a simplex. The discussion of the equality case is based on the equality case in the geometric form of the Brascamp-Lieb inequality. The present paper establishes stability versions of the reverse isoperimetric inequality and of the corresponding inequality for isotropic measures.
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Additional Information
  • Károly J. Böröczky
  • Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, 1053 Budapest, Reáltanoda u. 13-15, Hungary – and – Central European University, Nador utca 9, Budapest, H-1051 Hungary
  • Email: carlos@renyi.hu
  • Daniel Hug
  • Affiliation: Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany
  • MR Author ID: 363423
  • Email: daniel.hug@kit.edu
  • Received by editor(s): February 10, 2015
  • Received by editor(s) in revised form: October 2, 2015
  • Published electronically: March 1, 2017
  • Additional Notes: The first author was supported by NKFIH 109789 and 116451
    The second author was supported by DFG grants FOR 1548 and HU 1874/4-2
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 6987-7019
  • MSC (2010): Primary 52A40; Secondary 52A38, 52B12, 26D15
  • DOI: https://doi.org/10.1090/tran/6857
  • MathSciNet review: 3683100