Discrete Log-Concavity Under Self-Convolution

Definitions and mathematical statement

Let $d\geq 1$ be an integer. A probability mass function on ${\mathbb{Z}}^d$ is a function $p:{\mathbb{Z}}^d\to[0,1]$ satisfying $\sum_{k\in{\mathbb{Z}}^d}p(k)=1$. A function $f:{\mathbb{R}}^d\to[0,\infty)$ is log-concave if \[ f((1-t)x+ty)\geq f(x)^{1-t}f(y)^t \qquad(x,y\in{\mathbb{R}}^d,\ 0<t<1). \] Zeros are permitted; at $t=0,1$ the corresponding inequality is equality. Say that $p$ has a log-concave extension if there exists such an $f$ with $f(k)=p(k)$ for every $k\in{\mathbb{Z}}^d$. Define its self-convolution by \[ (p*p)(k)=\sum_{j\in{\mathbb{Z}}^d}p(j)p(k-j),\qquad k\in{\mathbb{Z}}^d. \] Question. For every positive integer $d$ and every probability mass function $p$ on ${\mathbb{Z}}^d$ having a log-concave extension, does there exist a log-concave function $g:{\mathbb{R}}^d\to[0,\infty)$ satisfying \[ g(k)=(p*p)(k)\qquad\text{for every }k\in{\mathbb{Z}}^d? \] No finite-support assumption, regularity condition, or Lebesgue-integral normalization of the extensions is specified in the supplied reconstructed statement, so none is imposed here. The positive-integer range for $d$ and the finite-valued definition above are conventional interpretations. The supplied original excerpt refers only to "this definition" without reproducing it; consequently, any further qualifications in that antecedent remain unresolved. This formulation follows the supplied reconstructed statement and does not assert a separate entropy inequality.

Short English statement

If a lattice probability mass function extends to a log-concave function on Euclidean space, must the distribution of the sum of two independent random vectors with that mass function also have such an extension? The supplied record labels this question open; the precise antecedent definition in the original excerpt requires review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006136$, source code catalogue entry; collection: Oberwolfach Reports Open Problems. Registry paths identified by the project are data/unsolvedmath/problems.json and the supplementary catalog/problems.json; their contents were not independently inspected for this draft.
• Supplied report citation: Convex Geometry and its Applications (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/58, pp. 3301--3376. https://doi.org/10.4171/owr/2024/58.
• The supplied original excerpt associates the question with Matthieu Fradelizi's contribution, On the monotonicity of discrete entropy for log-concave random vectors in ${\mathbb{Z}}^d$, described as joint work with Lampros Gavalakis and Martin Rapaport. The record does not identify a separate proposer.
• Supplied literature source: "OWR 2024/58, open problem." https://ems.press/content/serial-article-files/50771.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that the report leaves the preservation question unresolved, and reports finding no later proof or counterexample. This is supplied dated curation, not an independently verified literature assessment. The record also labels the reconstructed statement corrected_verified and says it was checked against the official report; that verification is supplied metadata, not a check performed for this draft.