[Paper Review] Functional versions of L_p-affine surface area and entropy inequalities

Umut Caglar, Matthieu Fradelizi+4 more|arXiv (Cornell University)|Feb 13, 2014

Point processes and geometric inequalities12 references4 citations

TL;DR

This paper introduces a functional extension of $L_p$-affine surface area for log-concave and $s$-concave functions, establishing duality relations and affine isoperimetric inequalities. It derives a new reverse log-Sobolev inequality for $s$-concave densities, generalizing classical results and recovering the log-Sobolev inequality in the limit as $s \to 0$. The approach uses functional forms of the Blaschke-Santaló inequality and Legendre duality, providing a unified framework for entropy and affine isoperimetric inequalities in the functional setting.

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.

Motivation & Objective

• To extend the concept of $L_p$-affine surface area from convex bodies to log-concave and $s$-concave functions.
• To establish functional duality relations and affine isoperimetric inequalities in the setting of $s$-concave and log-concave functions.
• To derive a new reverse log-Sobolev inequality for $s$-concave densities, generalizing known results.
• To provide a simplified proof of the reverse log-Sobolev inequality for log-concave measures using functional Blaschke-Santaló duality.

Proposed method

• Define a functional $L_p$-affine surface area via the Hessian determinant of the convex function $\psi$ associated with a log-concave or $s$-concave density $f = e^{-\psi}$.
• Use the Legendre transform $\psi^*$ and the functional Blaschke-Santaló inequality to derive duality relations between $f$ and its polar $f_{(s)}^\circ$.
• Apply reverse Hölder and Jensen inequalities to relate the entropy $S(\mu)$ and the integral of $\log \det(\nabla^2 \psi)$.
• Use the functional form of the Blaschke-Santaló inequality to bound the product $\int f\,dx \cdot \int f_{(s)}^\circ\,dx$ in terms of a Gaussian-type integral.
• Derive the reverse log-Sobolev inequality by combining the duality relation with entropy and Hessian determinant terms.
• Characterize equality in the inequality using the extremal case $f(x) = c_0(1 - s|Ax|^2)^{1/(2s)}$ for positive definite $A$.

Experimental results

Research questions

• RQ1 How can the $L_p$-affine surface area concept be generalized from convex bodies to log-concave and $s$-concave functions?
• RQ2 What duality relations hold between a function and its polar in the functional $L_p$-setting?
• RQ3 Can affine isoperimetric inequalities be established for $s$-concave functions, and how do they relate to classical inequalities?
• RQ4 What is the functional analog of the reverse log-Sobolev inequality for $s$-concave densities?
• RQ5 How does the new reverse log-Sobolev inequality recover the classical one as $s \to 0$?

Key findings

• A functional $L_p$-affine surface area is defined for $s$-concave functions via the Hessian determinant of the associated convex potential $\psi$.
• The functional Blaschke-Santaló inequality yields a duality relation between $f$ and its polar $f_{(s)}^\circ$, leading to a new reverse log-Sobolev inequality.
• The reverse log-Sobolev inequality for $s$-concave densities is given by $\int \log \det(\nabla^2 \psi)\, d\mu \leq \int \log\left(\left(1 + s(\langle x,\nabla\psi\rangle - \psi)\right)^{1/s + n}\right) d\mu - S(\mu) + \log\left(\left(\frac{\pi}{s}\right)^n \frac{(\Gamma(1+1/(2s)))^2}{(\Gamma(1+n/2+1/(2s)))^2}(1+ns)\right)$.
• Equality holds in the inequality if and only if $f(x) = c_0(1 - s|Ax|^2)^{1/(2s)}$ for a positive definite matrix $A$, with $c_0$ chosen so $\int f\,dx = 1$.
• As $s \to 0$, the reverse log-Sobolev inequality for $s$-concave functions converges to the classical reverse log-Sobolev inequality for log-concave measures.
• The new approach provides a short and general proof of the reverse log-Sobolev inequality, including the equality case, via functional duality and the Blaschke-Santaló inequality.