Necessary and sufficient conditions are given in order for a Borel measure on the Euclidean sphere to have an affine image that is isotropic. A sharp reverse affine isoperimetric inequality for Borel measures on the sphere is presented. This leads to sharp reverse affine isoperimetric inequalities for convex bodies.
"Affine images of isotropic measures." J. Differential Geom. 99 (3) 407 - 442, March 2015. https://doi.org/10.4310/jdg/1424880981