Ehrhart’s volume bound and equality cases for centered lattice bodies
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Ehrhart’s volume conjecture concerns a full-dimensional compact convex body in Rⁿ with barycenter zero and zero as its only interior lattice point. At volume (n+1)ⁿ/n!, must the body be an integral unimodular image of the centered simplex (n+1)Δₙ−(1,…,1)?
Why it matters and what counts as progress
Why it matters: An equality classification identifies every body attaining the proposed maximal volume, not merely the value of a bound.
What counts as progress: Prove the equality classification or establish related sharp volume bounds under weaker convex-geometric hypotheses.
How the problem developed
- 1964 · Conjecture [1]
Eugène Ehrhart Proposed the sharp upper bound for a centered convex body whose only interior lattice point is the origin; the equality refinement was later formulated explicitly by Nill and Paffenholz.
- 2014 · Partial result [2]
Benjamin Nill and Andreas Paffenholz Formulated the general equality conjecture and proved the inequality and equality classification for bodies contained in the polar of a lattice polytope.
- 1 Aug 2026 · Reported claim [3]
OpenAI internal model (Astra) The public report claims the sharp volume inequality in every dimension for centered bodies with the origin as their unique interior lattice point; it leaves the equality classification open.
- 24 Aug 2026 · Reported claim [1]
Jihao Liu, with GPT-5.6-sol, Fable 5 and Danus The manuscript classifies equality bodies in every dimension as unimodular images of the centered standard simplex. It credits generative AI, particularly the named systems, with the main result.
AI contributions 2
The equality case of Ehrhart's volume conjecture
Liu claims that a full-dimensional compact convex body in Rⁿ with barycenter zero, zero as its only interior lattice point and volume (n+1)ⁿ/n! is an integral unimodular image of the centered simplex (n+1)Δₙ−(1,…,1).
Ten Advances in Mathematics and Theoretical Computer Science
For every positive dimension n, a compact convex body with nonempty interior, barycenter 0 and no interior lattice point other than 0 has volume at most (n+1)ⁿ/n!. A centered simplex attains equality. This is the inequality, separate from the later equality classification.
Sources
- Liu, The equality case of Ehrhart’s volume conjecture
- Nill and Paffenholz, On the equality case in Ehrhart’s volume conjecture
- OpenAI, Ten advances in mathematics and theoretical computer science
- OpenAI, Ten advances: research account
- OpenAI ten-proofs formalization manifest
- Lean proof of the Ehrhart volume inequality
- Sienicki et al., chapter-by-chapter mathematical review
- Liu, separate Ehrhart equality classification
Website timeline
Publication dates and editorial changes.
Danus-assisted proof characterizes equality in Ehrhart’s volume bound