ISOTROPIC MEASURES AND STRONGER FORMS OF THE REVERSE ISOPERIMETRIC INEQUALITY

摘要

The reverse isoperimetric inequality, due to Keith Ball, states that if K is an n-dimensional convex body, then there is an affine image (K) over tilde of K for which S((K) over tilde)(n)/V ((K) over tilde)(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.