Sharp Rogers--Shephard Inequalities for Rényi Entropy
Abstract
We establish sharp weighted Rogers--Shephard inequalities for Rényi differential entropy. For independent copies X,X' of a log-concave random vector in Rn, we prove hp(aX−bX')≤ hp(X)+nlog(a+b) for all a,b>0 and 1≤ p≤∞, where orders $1$ and ∞ 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≤ p<∞, equality holds precisely for invertible affine images of random vectors with independent exponential coordinates. At p=∞, 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.
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BibTeX
@misc{han2026sharp,
title = {{Sharp Rogers--Shephard Inequalities for Rényi Entropy}},
author = {Bang-Xian Han and Guoqing Zhan and Chiyu Zhou and Zhuo-Nan Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15716},
eprint = {2610.15716},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15716},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Sharp Rogers--Shephard Inequalities for Rényi Entropy
AU - Bang-Xian Han
AU - Guoqing Zhan
AU - Chiyu Zhou
AU - Zhuo-Nan Zhu
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15716
UR - https://arxiv.org/abs/2610.15716
AB - We establish sharp weighted Rogers--Shephard inequalities for Rényi differential entropy. For independent copies X,X' of a log-concave random vector in R^n, we prove h_p(aX−bX')≤ h_p(X)+nlog(a+b) for all a,b>0 and 1≤ p≤∞, where orders $1$ and ∞ 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≤ p<∞, equality holds precisely for invertible affine images of random vectors with independent exponential coordinates. At p=∞, 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.
ER -