Functional versions of L_p-affine surface area and entropy inequalities

2014/02/13 by Umut Caglar, U. Caglar, M. Fradelizi +15 · 4 citations
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Markov Chains and Monte Carlo Methods #Mathematical Inequalities and Applications #Point processes and geometric inequalities #math.FA

paper · pdf · doi:10.48550/arxiv.1402.3250

arxiv created 2014/02/13 · openalex publication_date 2014/02/13 · arxiv updated 2014/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In contemporary convex geometry, the rapidly developing L_p-Brunn Minkowski theory is a modern analogue of the classical Brunn Minkowski theory. A cornerstone of this theory is the L_p-affine surface area for convex bodies. Here, we introduce a functional form of this concept, for log concave and s-concave functions. We show that the new functional form is a generalization of the original L_p-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.

Citations

• Mixed f-divergence and inequalities for log concave functions

Cited by

• A Characterization of Functional Affine Surface Areas
• On the Semicontinuity of Functionals on Function Spaces
• Grünbaum's inequality for Gaussian and convex probability measures
• A Minkowski problem for α-concave functions via optimal transport

Related

• Geometry of log-concave functions: the L_p Asplund sum and the L_p Minkowski problem
• The area measure of log-concave functions and related inequalities
• Pinsker inequalities and related Monge-Ampère equations for log concave functions
• Affine isoperimetric inequalities in the functional Orlicz-Brunn-Minkowski theory
• Surface area measures of log-concave functions