Sharp Rogers--Shephard Inequalities for Rényi Entropy

Abstract

We establish sharp weighted Rogers--Shephard inequalities for Rényi differential entropy. For independent copies $X,X^{\prime }$ of a log-concave random vector in ${\mathbb{R}}^n$, we prove $h_p(aX-b{X}^{\prime })\le h_p(X)+n\log (a+b)$ for all $a,b>0$ and $1\le p\le \infty$, where orders $1$ and $\infty$ correspond to Shannon entropy and min-entropy, respectively. The Shannon case settles the entropic Rogers--Shephard conjecture of Madiman and Kontoyiannis and its higher-dimensional formulation. For $1\le p<\infty$, equality holds precisely for invertible affine images of random vectors with independent exponential coordinates. At $p=\infty$, the larger equality class is characterized by possibly extended-valued convex potentials that are positively homogeneous of degree one. For every $0<p<1$, uniform laws on sufficiently high-dimensional simplices violate the unweighted bound. Thus $p=1$ is the critical order for this bound to hold in every dimension. The proof combines weighted convex intersections with an exponential lift to a log-affine probability density. At the Shannon endpoint, this lift yields an exact deficit decomposition with a conditional mutual information term for an additive channel with asymmetric Laplace noise.

arXiv comment29 pages; comments are welcome!