Let n ≥ 1 and let Z^n ⊂ R^n be the standard lattice. For a convex body K ⊂ R^n (a compact convex set with nonempty interior) write operatornameVol(K) for its n-dimensional Lebesgue measure and c(K) ;=; frac1operatornameVol(K)int_K x , dx for its barycenter. Call two subsets K, K' ⊂ R^n unimodularly equivalent if K' = A K + b for some A ∈ GL_n(Z) and b ∈ Z^n. Let (n+1)Delta_n ;=; operatornameconvBig-∑_i=1^n mathbfe_i, (n+1)mathbfe_j-∑_i=1^n mathbfe_i : j=1,…,n Big, where mathbfe_1,…,mathbfe_n is the standard basis of R^n; its unique interior lattice point is the origin, which is also its barycenter. Suppose K ⊂ R^n is a convex body such that operatornameint(K) cap Z^n = c(K), i.e. K contains exactly one lattice point in its interior and that point is the barycenter of K. Then operatornameVol(K) ;≤; frac(n+1)^nn!, and equality holds if and only if K is unimodularly equivalent to (n+1)Delta_n.

Ehrhart's volume conjecture

About 62 years old · traced to 1964

Let $n \ge 1$ and let $\mathbb{Z}^n \subset \mathbb{R}^n$ be the standard lattice. For a convex body $K \subset \mathbb{R}^n$ (a compact convex set with nonempty interior) write $\operatorname{Vol}(K)$ for its $n$ -dimensional Lebesgue measure and

c(K) \;=\; \frac{1}{\operatorname{Vol}(K)}\int_K x \, dx

for its barycenter. Call two subsets $K, K' \subset \mathbb{R}^n$ unimodularly equivalent if $K' = A K + b$ for some $A \in GL_n(\mathbb{Z})$ and $b \in \mathbb{Z}^n$ . Let

(n+1)\Delta_n \;=\; \operatorname{conv}\Big\{-\sum_{i=1}^n \mathbf{e}_i,\ (n+1)\mathbf{e}_j-\sum_{i=1}^n \mathbf{e}_i \ :\ j=1,\dots,n\Big\},

where $\mathbf{e}_1,\dots,\mathbf{e}_n$ is the standard basis of $\mathbb{R}^n$ ; its unique interior lattice point is the origin, which is also its barycenter.

Suppose $K \subset \mathbb{R}^n$ is a convex body such that $\operatorname{int}(K) \cap \mathbb{Z}^n = \{c(K)\}$ , i.e. $K$ contains exactly one lattice point in its interior and that point is the barycenter of $K$ . Then

\operatorname{Vol}(K) \;\le\; \frac{(n+1)^n}{n!},

and equality holds if and only if $K$ is unimodularly equivalent to $(n+1)\Delta_n$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Ehrhart’s sharp volume conjecture
Does the centered simplex maximize volume among convex bodies whose barycenter is their unique interior lattice point?

References

Primary source

Wikipedia

Additional references

Wikipedia, Ehrhart's volume conjecture, the article this problem comes from.

geometry unsolved-problem

Posted 2 months ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 preprint claims the conjecture is solved in every dimension, but its proof and the separately reported bound have not yet been independently verified.

Ehrhart’s 1964 conjecture says that a convex body whose barycenter is its only interior lattice point has volume at most that of the centered standard simplex, with equality only for a unimodular copy of that simplex.

Known results

Ehrhart proved the upper bound in dimension $2$ and for simplices in every dimension (1964).
Equality-case uniqueness was proved in dimension $2$ .
Berman and Berndtsson established the conjectured bound for a class including reflexive polytopes (2012).
Nill and Paffenholz formulated the general equality refinement as Conjecture 1.1 (2014).

August 2026 claimed resolution

A preprint claims the equality case in every dimension: equality forces $K$ to be a unimodular image of the centered simplex. It says the inequality was recently proved by OpenAI, but gives no identified publication for that proof. The preprint attributes its equality result to GPT-5.6-sol, Fable 5, and the Danus system; these claims remain unverified.

Current status (as of August 2026): The classical cases are settled, and a 2026 preprint claims both the general inequality and equality characterization, but the full conjecture remains unverified.

Sources

No solutions have been posted yet.

Ehrhart’s sharp volume conjecture

Primary source

Additional references

Progress summary