Isotropic measures and stronger forms of the reverse isoperimetric inequality

Böröczky, Károly; Hug, Daniel

doi:10.1090/tran/6857

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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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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