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The Equality Case of Ehrhart's Volume Conjecture

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ehrhart-volume-conjecture-equalityGeometry & topologyposed by Eugene Ehrhart, 1964recorded: solved

1 attempt · no person has looked

Statement

Ehrhart conjectured that a full-dimensional compact convex body in Rn\mathbb{R}^n whose barycenter is its unique interior lattice point has volume at most (n+1)n/n!(n+1)^n/n!. With the inequality itself settled, the remaining question was which bodies attain it. Every such body is a unimodular image of the simplex (n+1)Δn−(1,…,1)(n+1)\Delta_n - (1,\dots,1).

Context

the equality case; the inequality was settled separately and is tracked on its own entry

Completes Ehrhart's 1964 conjecture. The inequality was proved shortly before this, and this is the characterisation of the bodies attaining the bound, which the inequality alone leaves open.

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1 attempt

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  • #1

    Attempt 1

    proof attemptGPT-5.6 Sol, Fable 5, Danus with Jihao Liu ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.6 Sol, Fable 5, Danus
    people
    Jihao Liu

    The paper states the main result was obtained by generative AI, naming GPT-5.6-sol, Fable 5 and the Danus system, and its comment records essential human strategic input followed by human verification.

    the equality case; the inequality was settled separately and is tracked on its own entry

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