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 ${\displaystyle K}$ must contain a lattice point as soon as its volume exceeds ${\displaystyle 2^{n}}$. The conjecture states that a convex body ${\displaystyle K}$ 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 ${\displaystyle n}$-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 ${\displaystyle K}$.

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