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

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 $(\log f)&#39;&#39;&lt;-1/2$, for which the corresponding function satisfies $F&#39;&#39;(t)&gt;0$ throughout an endpoint neighborhood $0<t<δ$. This disproves the conjecture without an additional symmetry assumption. For a class of Gaussian perturbation densities, we first establish the endpoint expansion \[ F''(t)=-\frac{μ_3J_3}{16\sqrt t}+O(1),\qquad t\downarrow0. \] Here $μ_3=\int x^3f(x)\,dx$ is the third central moment. Writing $ρ=(\log f)&#39;$ for the score function, its third moment is $J_3=\int f(x)ρ(x)^3\,dx$. A Hermite perturbation gives $$μ_3\&gt;0$$ and $J_3&lt;0$, and explicit remainder estimates verify the constructed density and its endpoint curvature. The counterexample does not address the conjecture with an additional symmetry assumption.