Isotropic measures and stronger forms of the reverse isoperimetric inequality

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.