The equality case of Ehrhart's volume conjecture

Abstract

We prove that every full-dimensional compact convex body in ${\mathbb{R}}^n$ whose barycenter is its unique interior lattice point and whose volume is $(n+1{)}^n/n!$ is a unimodular image of the simplex $(n+1){\Delta }_n-(1,\dots ,1)$. This resolves the equality case of Ehrhart's volume conjecture, as a counterpart of the inequality part recently proved by OpenAI. The main result of this paper is obtained by generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system.

30 pages. AI generated with essential human strategical idea input. Human verified. Version 2, no math change, references updated