Let X and Y be independent identically distributed real-valued random variables with a common log-concave density and finite differential entropy. Define F:[0,1]→R by F(t)=h bigl(√1-t,X+√t,Y bigr). The Ball–Nayar–Tkocz entropy concavity conjecture asserts that F is concave on [0,1]; equivalently, for all s,t∈[0,1] and lambda∈[0,1], F bigl((1-lambda)s+lambda t bigr)≥ (1-lambda)F(s)+lambda F(t).

Ball–Nayar–Tkocz entropy concavity conjecture

Let $X$ and $Y$ be independent identically distributed real-valued random variables with a common log-concave density and finite differential entropy. Define $F:[0,1]\to\mathbb{R}$ by $F(t)=h\bigl(\sqrt{1-t}\,X+\sqrt{t}\,Y\bigr)$ . The Ball–Nayar–Tkocz entropy concavity conjecture asserts that $F$ is concave on $[0,1]$ ; equivalently, for all $s,t\in[0,1]$ and $\lambda\in[0,1]$ , $F\bigl((1-\lambda)s+\lambda t\bigr)\ge (1-\lambda)F(s)+\lambda F(t)$ .

References

Primary source

arXiv

Additional references

Entropy concavity for log-concave random variables: an asymmetric counterexample — arXiv — Congyi Luo

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed progress

A 2026 preprint gives a counterexample when the variables are not symmetric, while the symmetric case remains open.

Ball, Nayar, and Tkocz conjectured in 2015 that entropy along normalized sums of independent identically distributed log-concave variables is concave. The new result addresses the original nonsymmetric formulation, not its symmetry-restricted variant.

Known results

The 2015 source records the conjecture and proves only a weakened related projection-entropy inequality, with constant $\kappa=1/5$ rather than the conjectured $\kappa=1$ .

September 2026 asymmetric counterexample

Congyi Luo’s preprint Entropy concavity for log-concave random variables: an asymmetric counterexample reports a counterexample to the conjecture without symmetry. It therefore refutes the unrestricted formulation if confirmed, while leaving the symmetric log-concave case open; the claim has not been independently verified here.

Current status (as of September 2026): the unrestricted conjecture is claimed refuted by an unverified counterexample, while the symmetry-restricted version remains open.

Sources

No solutions have been posted yet.