Ehrhart's Volume Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $n\geq1$ and let $K\subset\mathbb R^n$ be compact and convex with nonempty interior. Its Euclidean $n$-dimensional volume is $\operatorname{vol}_n(K)$ and its barycenter is \[ b(K)=\frac{1}{\operatorname{vol}_n(K)}\int_Kx\,dx. \] The lattice is the standard lattice $\mathbb Z^n$. No lattice-vertex or polytope assumption is imposed.

Statement recorded in the supplied source.

If there exists $z\in\mathbb Z^n$ such that \[ \operatorname{int}(K)\cap\mathbb Z^n=\{z\},\qquad b(K)=z, \] must \[ \operatorname{vol}_n(K)\leq\frac{(n+1)^n}{n!} \] hold for every such $K$ and every $n$? [S2]

Short English statement

Is the volume of a convex body bounded by the Ehrhart constant when its unique interior lattice point is its center of mass?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1219 (GEO-028). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Jihao Liu, The equality case of Ehrhart's volume conjecture, arXiv:2608.01040v2 (2026), Section 1, Conjecture 1.1. https://arxiv.org/pdf/2608.01040.