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Ehrhart's Volume Conjecture

What is the maximum volume of a convex body in Rn\mathbb{R}^n whose centroid is its only interior lattice point? Ehrhart conjectured the extremal value in 1964; the sharp maximum is now determined in every dimension.

Result
Proved
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Convex geometry
Posed by
Eugène Ehrhart
Year posed
1964
Years open
62y
Solved
2026-08-01
Model
Astra (internal preview)
Vendor
OpenAI
Collaborators
—
Verification
Lean-verified
Publication
Announced
Significance
25 / 100
Disclosed cost
$182
Wikipedia
1 language

What the AI did

Generated by an internal version of OpenAI's Astra: per the announcement, the mathematical arguments were produced by the system (roughly 2,000 dollars of compute at Sol API rates across all ten results), humans prepared the manuscripts with the same model, and the model then formalized the argument in Lean. A narrated reasoning walkthrough is published for each result.

Verification

Kernel-checked Lean 4 certificate in OpenAI's public ten-proofs repository (Lean 4.32, mathlib, `lake build All`), with an independent Comparator checking route. Statement fidelity and community review of the day-old company announcement remain pending, hence candidate status.

Sources

Changelog4 changes
  • Rasmus Lindahlcommented
  • Rasmus Lindahlchanged Wikipedia from “No dedicated article” to “1 language”
  • VelvetPuffin429commented
  • VelvetPuffin429set Cost note to “Not disclosed per result. OpenAI's announcement gives roughly $2,000 of compute at API rat…”, also More links, Disclosed cost (USD), Disclosed cost (USD), More links, Cost note

Discussion2

Rasmus Lindahl02 Aug 2026, 17:59 UTC· edited

Good catch, thanks. Fixed: the Wikipedia field now reads 1 language.

The article has existed since May 2016, so it predates the solution by a decade and counts under our freeze rule. Coverage created in response to a result does not count, which is why the field is a frozen snapshot rather than a live number.

We missed it for a duller reason. This entry was added after our last notability sweep, and the count defaults to 0, which the entry page then renders as "No dedicated article". That is a claim we had not actually earned here.

I re-ran the check across all 252 entries after your comment, and this was the only one affected. The sweep is a script now, so it can be re-run instead of relying on the default.

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