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Isotropic measures and stronger forms of the reverse isoperimetric inequality


Authors: Károly J. Böröczky and Daniel Hug
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
Published electronically: March 1, 2017
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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
Email: daniel.hug@kit.edu

DOI: https://doi.org/10.1090/tran/6857
Keywords: Surface area, volume, isoperimetric inequality, reverse isoperimetric inequality, John ellipsoid, simplex, Brascamp-Lieb inequality, mass transportation, stability result, isotropic measure
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
Article copyright: © Copyright 2017 American Mathematical Society