# How large can a convex body with one interior lattice point be?

- ID: ehrhart-volume-conjecture
- Provider: OpenAI
- Published: 2026-08-01
- Editorial check: 2026-10-29
- Status: formalized-provider-reported
- Evidence: Provider announcement; formal certificates reported, not rerun here
- Source: https://openai.com/index/ten-advances-in-mathematics/
- Public article: https://appliedfrontier.nl/en/discoveries/ehrhart-volume-conjecture/

## Plain-language explanation

Take a solid shape without dents and place it on an integer grid. Suppose its center of mass is the only grid point strictly inside. How much volume can the shape have?

## Research contribution

The August release reports a resolution of Ehrhart’s volume conjecture in every dimension for the stated centroid condition.

## Academic interpretation

The centroid condition carries real mathematical content: merely requiring one interior lattice point is a different problem. One must also specify the ambient lattice and the normalization of volume. A robust verification checks that the formal statement uses the same interior, centroid and convexity assumptions as the natural-language theorem.

## Implications

Potentially connects discrete lattice constraints to continuous geometric size, with implications for the geometry of numbers.

## Caveat

The result applies to a precisely constrained class of convex bodies. This page has not checked the formal certificate or classified equality cases.

This is an editorial summary, NOT an independent mathematical proof check.