Affine images of isotropic measures

Abstract

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.

Cone-volume measure of general centered convex bodies

“…In [12, Theorem 1.3] an optimal lower bound on the U-functional of a measure satisfying the subspace concentration condition is given. In combination with Theorem 1.1 we immediately get the best possible bound on U(K) in terms of V(K) which was conjectured in [12].…”

“…Böröczky, E. Lutwak, D. Yang and G. Zhang [12], while the discrete case was established earlier by E.A. Carlen and D. Cordero-Erausquin [14], and J. Bennett, A. Carbery, M. Christ and T. Tao [7] in their study of the Brascamp-Lieb inequality.…”

“…Böröczky, E. Lutwak, D. Yang and G. Zhang [12]. In [12, Theorem 1.3] an optimal lower bound on the U-functional of a measure satisfying the subspace concentration condition is given.…”

On the Discrete Logarithmic Minkowski Problem

“…For more references on the progress of the subspace concentration condition, see, e.g., Henk et al [32], He et al [30], Xiong [67], Böröczky et al [8], and Henk and Linke [31].…”

Sharp affine isoperimetric inequalities for the Minkowski first mixed volume

“…In 2015, Böröczky and LYZ extended the domain of functional from the class of polytopes to the set of convex bodies in with origin in their interiors, , and defined Since , it follows that . It is still centro‐affine invariant, that is, , for .…”