Entropy concavity for log-concave random variables: an asymmetric counterexample

Congyi Luo

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arXiv · 2609.11418

Entropy concavity for log-concave random variables: an asymmetric counterexample

Congyi Luo

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Abstract

The Ball-Nayar-Tkocz entropy concavity conjecture asserts that, if $X,Y$ are independent identically distributed real random variables with a common log-concave density, then the differential entropy of their weighted sum, \[ F(t)=h\bigl(\sqrt{1-t}\,X+\sqrt t\,Y\bigr),\qquad 0\le t\le1, \] is a concave function of the weight parameter $t$. We construct an asymmetric, strictly positive smooth probability density $f$ with mean zero, variance one, and $(\log f)''<-1/2$, for which the corresponding function satisfies $F''(t)>0$ throughout an endpoint neighborhood $0 0$ and $J_3<0$, and explicit remainder estimates verify the constructed density and its endpoint curvature. The counterexample does not address the conjecture with an additional symmetry assumption.

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Congyi Luo. 2026-09-10. Entropy concavity for log-concave random variables: an asymmetric counterexample. https://arxiv.org/abs/2609.11418

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