Cone-volume measure of general centered convex bodies

The Orlicz version of the L Minkowski problem for −n < p < 0

“…In turn, we conclude (10). It follows from (10) and Lemma 3.2 that for the L 1 function f =f n n+q , we have…”

The Gauss Image Problem

“…A good way to do that is to contrast the Gauss image problem with a specific Minkowski problem, say the log-Minkowski problem. The cone volume measure of a convex body has been of considerable recent interest (see, e.g., [9,12,13,16,25,26,43,44,54]). The conevolume measure V K of a convex body K is a Borel measure on the unit sphere, defined for Borel !…”

Strengthened volume inequalities for $L_p$ zonoids of even isotropic measures

“…Even if we do not use it in this paper, we point out that F. Barthe [7] proved "continuous" versions of the Brascamp-Lieb and the reverse Brascamp-Lieb inequalities that work for any isotropic measure µ on S n−1 (see (12) and (13) below). Here we only consider the case in which all non-negative real functions involved coincide with a "nice" probability density function, which is the common case in geometric applications.…”