Log-concavity at the first three interior Ehrhart values

Determine whether every very ample lattice polytope satisfies the inequality |E_P(-2)|^2<|E_P(-1)||E_P(-3)| for its interior Ehrhart polynomial values.

Background

The paper constructs a Gorenstein lattice polytope with a regular unimodular flag triangulation whose interior Ehrhart series is not log-concave, thereby answering part (b) of Question 5.12 affirmatively. However, the authors distinguish part (a), which asks for a failure of log-concavity specifically at the first three values |E_P(-1)|, |E_P(-2)|, and |E_P(-3)|. Their constructed example cannot address this case because all three of these values vanish, so the existence of a very ample polytope satisfying the strict reverse inequality at these indices remains unresolved in the paper.

References

Part~(a) asks specifically for $|E_P(-2)|2<|E_P(-1)||E_P(-3)|$; the preceding example does not settle it, since $|E_R(-1)|=|E_R(-2)|=|E_R(-3)|=0$.

— Unimodality shenanigans in Ehrhart theory  (2609.10513 - Ferroni, 9 Sep 2026) in Section 5, immediately after Corollary 5.2 (discussion of Question 5.12(b))