Math Conjectures

Ehrhart Volume Conjecture

OPENMajorConjectureProposed 1964 · Full conjecture

Canonical statement

Let K⊂RnK\subset\mathbb R^n be a convex body whose barycenter is the origin and whose only interior lattice point is 00. Then
vol⁡(K)≤(n+1)nn!. \operatorname{vol}(K)\le \frac{(n+1)^n}{n!}.
Moreover, equality holds only for unimodular images of the centered simplex
(n+1)conv⁡(0,e1,…,en)−(1,…,1). (n+1)\operatorname{conv}(0,e_1,\ldots,e_n)-(1,\ldots,1).
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Let \(K\subset\mathbb R^n\) be a convex body whose barycenter is the origin and whose only interior lattice point is \(0\). Then
\[
  \operatorname{vol}(K)\le \frac{(n+1)^n}{n!}.
\]
Moreover, equality holds only for unimodular images of the centered simplex
\[
  (n+1)\operatorname{conv}(0,e_1,\ldots,e_n)-(1,\ldots,1).
\]
DG-019:[Ehrhart1964Volume][NillPaffenholz2014Ehrhart][OpenAI2026TenAdvances]
Discrete geometryGeometry and topology

Notes

Ehrhart conjectured the sharp volume bound for a centered convex body with a unique interior lattice point [Ehrhart1964Volume]. The equality case predicts the centered simplex (n+1)conv⁡(0,e1,…,en)−(1,…,1)(n+1)\operatorname{conv}(0,e_1,\ldots,e_n)-(1,\ldots,1), and its classification is an integral part of the full form [NillPaffenholz2014Ehrhart]. The August 1, 2026 manuscript claims the inequality but explicitly leaves equality open [OpenAI2026TenAdvances], so even a verified proof would update rather than completely close this record.

Proof-claim watch (1)

Ehrhart sharp volume bound

Grade C

Sharp-inequality claim pending verification; the full catalog record remains open because equality is not classified.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.