## Isotropic measures and stronger forms of the reverse isoperimetric inequality

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- by Károly J. Böröczky and Daniel Hug PDF
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**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.## References

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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