Isotropic measures and stronger forms of the reverse isoperimetric inequality

The reverse isoperimetric inequality, due to Keith Ball, states that if K K is an n n -dimensional convex body, then there is an affine image K ~ K̃ of K K for which S ( K ~ ) n / V ( K ~ ) n − 1 S(K̃)^n/V(K̃)ⁿ⁻¹ is bounded from above by the corresponding expression for a regular n n -dimensional simplex, where S S and V V denote the surface area and volume functional. It was shown by Franck Barthe that the upper bound is attained only if K 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.