An AI-generated proof given by OpenAI (2026) established this bound in every dimension.
The bound is sharp. If ,
then the centered simplex has centroid 0, contains no other interior
lattice point, and has volume .
The conjectured classification of equality cases states that every extremizer is
the image of this simplex under a unimodular matrix.
This classification remains open (Nill and Paffenholz 2014).
Ehrhart, E. "Une généralisation probable du théorème fondamental de Minkowski." C. R. Acad. Sci.
Paris258, 4885-4887, 1964.Nill, B. and Paffenholz, A. "On
the Equality Case in Ehrhart's Volume Conjecture." Adv. Geom.14,
579-586, 2014.OpenAI. "The Sharp Inequality in Ehrhart's Volume
Conjecture." Ch. 8 in Ten Advances in Mathematics and Theoretical Computer
Science. Aug. 1, 2026. https://cdn.openai.com/pdf/ten-proofs-oai.pdf.
is a full-dimensional compact convex body with centroid
0 whose only interior lattice point is the origin,
so
,
then the centered simplex 
has centroid 0, contains no other interior
lattice point, and has volume 
.
The conjectured classification of equality cases states that every extremizer is
the image of this simplex under a unimodular matrix.
This classification remains open (Nill and Paffenholz 2014).
