60-second version
Take a convex body—something with no inward dents—and place it around a lattice of integer points. Suppose its centroid is the only lattice point in its interior. How large can the body be?
OpenAI reports the sharp answer in every dimension: the maximum volume is (n+1)^n/n!, with equality at a simplex. The theorem identifies an extremal shape at the exact moment before another interior lattice point must appear.
Start with a triangle
In two dimensions, a triangle is the simplest convex body with three corners. Put the integer lattice underneath it and move the edges outward. The area grows until a new lattice point slips into the interior. The question is whether a different shape could delay that event while enclosing more area.
A convex body is allowed to have curved or many-sided boundaries, but it cannot have a dent. The centroid—the average position of all the body’s mass—must itself be controlled by the lattice condition. The surprising part is that the extremal answer is a simplex, the shape that seems least symmetric among the basic convex bodies.
Definitions that matter
A lattice is a discrete additive grid, usually ℤ^n. A convex body is compact, convex, and has nonempty interior. Its barycenter or centroid is the average of its points with respect to volume. The hypothesis says the centroid is the only lattice point strictly inside the body.
A sharp bound gives both an inequality and an equality case. Here the inequality says every allowed body has volume at most (n+1)^n/n!, while the equality case says a simplex can attain that volume. “Sharp” means the constant cannot be improved.
The research question
How much volume can a convex body contain if its centroid is its only interior lattice point? Does the maximum depend on the body’s detailed shape, or is there a universal formula in each dimension?
Ehrhart’s conjecture predicted the universal answer. Proving it means controlling continuous volume with a discrete condition that can change when a boundary crosses a single lattice point.
What was known before
The geometry of numbers has many theorems that turn lattice-point scarcity into volume bounds. Earlier results bounded the size of convex bodies with few interior lattice points, and simplex examples repeatedly appeared near extremality. But a general sharp formula tied specifically to the centroid condition remained conjectural.
The difficulty was not just finding a candidate. A proof had to show that every convex body, including irregular ones, can be compared with a shape whose volume is controlled by the same lattice-point information.
Why previous approaches stalled
The centroid is a global statistic. Moving one far-away vertex can shift it while barely changing another part of the boundary, and convex bodies need not have a convenient coordinate description. Meanwhile, the condition about interior lattice points is discontinuous: one lattice point crossing the boundary changes the hypothesis.
A successful argument must combine continuous geometry, affine transformations, and discrete counting without losing the equality case. The extremal simplex is not just a convenient picture; it is the configuration in which all of those inequalities line up.
The new result
The manuscript proves the sharp bound (n+1)^n/n! in every dimension for a convex body whose barycenter is its only interior lattice point. A simplex attains equality.
The visual shows expansion as a controlled experiment. Before another interior point appears, the simplex reaches the maximum allowed volume. Once a new lattice point enters, the premise has changed and the body is no longer an extremizer for the stated problem.
Formal result and proof architecture
The proof reduces the convex-body problem to a simplex inequality using barycentric coordinates and lattice-point constraints. The equality analysis tracks when each intermediate bound is tight, forcing the extremal configuration to be a simplex up to the relevant affine equivalences.
The Lean certificate encodes the finite-dimensional inequalities and equality conditions supported by the formal development. The manuscript contains the geometric definitions, the normalization conventions, and the route from centroid uniqueness to the volume formula.
What it could lead to
Sharp convex-body inequalities become reusable tools. The theorem can serve as a comparison principle in discrete geometry, where one wants to replace an arbitrary body by an extremal simplex and preserve enough information to bound volume or lattice counts.
For an AI research system, the result highlights a recurring pattern: a conjecture can be cracked by identifying the right equality case and then proving that every deviation spends some of the available geometric budget. That is a mathematical strategy, not a general shape optimizer.
What not to conclude
The formula does not say every large convex body resembles a simplex, and it does not solve all lattice-point volume problems. It applies to a specific centroid and interior-lattice-point condition.