Global log-concavity of unimodular fiber counts

Establish whether, for every unimodular matrix M and every probability distribution μ on the integer lattice with integer mean b, the number of nonnegative integer solutions to My = b is at least the μ-weighted geometric mean of the corresponding fiber cardinalities.

Background

The paper proves midpoint log-concavity for lattice-point counts in fibers of unimodular systems, including a coefficient-wise strengthening for bounded fibers. It explains that this establishes log-concavity along integer lines but does not imply global convexity or log-concavity on the full integer lattice.

The unresolved question asks whether the midpoint result extends to arbitrary probability distributions on \mathbb{Z}m, possibly with finite support. A positive answer would potentially enable stronger, possibly simply exponential, capacity bounds by allowing capacity arguments to use geometric directions adapted to the Newton polyhedron rather than only coordinate directions.

References

This theorem raises an open question, which is the natural generalization of Barvinok's question in . Let $M \in Z{m \times n}$ be a unimodular matrix and $\mu$ a probability distribution on $Zm$ (perhaps with finite support) such that its mean is $\bm{b} \in Zm$. Is it that case that

#{\bm{y} \in Z_{\geq 0}n \mid M\bm{y} = \bm{b}} \ge \prod_{\bm{a} \in Zm} #{\bm{y} \in Z_{\geq 0}n \mid M\bm{y} = \bm{a}}{\mu(\bm{a})}?

— Log-concavity and Approximate Counting for Totally Unimodular Polytopes  (2609.39917 - Leake et al., 30 Sep 2026) in Section 'An Open Question', Question \Cref{open-question-LC}