Functional versions of \(L_{p}\)-affine surface area and entropy inequalities (Q2797857)

Analytic versions of several important geometric invariants are defined and employed for deriving new inequalities. Among them, the most important is a new functional analog of the concept of \(L^p\)-affine surface areas of convex bodies that is introduced for \(\log\)-concave and \(s\)-concave functions, respectively. Duality relations and affine isoperimetric inequalities involving the latter are then provided, leading also to a new inverse \(\log\)-Sobolev inequality for \(s\)-concave densities, which thus provides a positive answer to an open question formulated in [\textit{S. Artstein-Avidan} et al., J. Funct. Anal. 262, No. 9, 4181--4204 (2012; Zbl 1255.52010)] by some of the authors.