Ehrhart's volume conjecture

In the geometry of numbers, Ehrhart's volume conjecture gives an upper bound on the volume of a convex body containing only one lattice point in its interior. It is a kind of converse to Minkowski's theorem, which guarantees that a centrally symmetric convex body K must contain a lattice point as soon as its volume exceeds ${\displaystyle 2^{n}}$. The conjecture states that a convex body K containing only one lattice point in its interior as its barycenter cannot have volume greater than ${\displaystyle (n+1)^{n}/n!}$:

${\displaystyle \operatorname {Vol} (K)\leq {\frac {(n+1)^{n}}{n!}}.}$

Equality is achieved in this inequality when ${\displaystyle K=(n+1)\Delta _{n}}$ is a copy of the standard simplex in Euclidean n-dimensional space, whose sides are scaled up by a factor of ${\displaystyle n+1}$. Equivalently, ${\displaystyle K=(n+1)\Delta _{n}}$ is congruent to the convex hull of the vectors ${\displaystyle -\sum _{i=1}^{n}\mathbf {e} _{i}}$, and ${\displaystyle (n+1)\mathbf {e} _{j}-\sum _{i=1}^{n}\mathbf {e} _{i}}$ for all ${\displaystyle j=1,\ldots ,n}$. Presented in this manner, the origin is the only lattice point interior to the convex body K.

The conjecture, furthermore, asserts that equality is achieved in the above inequality if and only if K is unimodularly equivalent to ${\displaystyle (n+1)\Delta _{n}}$.

Ehrhart proved the conjecture in dimension 2 and in the case of simplices.