OpenAlex ·
Near-Gaussian counterexamples to the Ball-Nayar-Tkocz entropy concavity conjecture
- Published
- Source
- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2609.11418
- Original
- Read it at the publisher
- Topics
Abstract
The Ball-Nayar-Tkocz conjecture asserts that, for independent real random variables $X,Y$ with a common log-concave density, the function $t\mapsto h(\sqrt tX+\sqrt{1-t}Y)$ is concave on $[0,1]$. Gaussian distributions have this property. We construct strongly log-concave counterexamples arbitrarily close to the Gaussian and study their persistence under Gaussian smoothing. Let $X$ have mean zero and variance one, let $G$ be an independent standard Gaussian, and let $q\in[0,1]$ denote the signal variance proportion in the smoothed variable $\sqrt qX+\sqrt{1-q}G$. In the asymmetric case, for every fixed $0
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