Functional Versions of L p -Affine Surface Area and Entropy Inequalities

In contemporary convex geometry, the rapidly developing |Lₚ|-Brunn–Minkowski theory is a modern analog of the classical Brunn–Minkowski theory. A central notion of this theory is the |Lₚ|-affine surface area of convex bodies. Here, we introduce a functional analog of this concept, for log-concave and |s|-concave functions. We show that the new analytic notion is a generalization of the original |Lₚ|-affine surface area. We prove duality relations and affine isoperimetric inequalities for log-concave and |s|-concave functions. This leads to a new inverse log-Sobolev inequality for |s|-concave densities.