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 $2^{n}$. The conjecture states that a convex body $K$ containing only one lattice point in its interior as its barycenter cannot have volume greater than $(n+1)^{n}/n!$: $\operatorname {Vol} (K)\leq {\frac {(n+1)^{n}}{n!}}.$ Equality is achieved in this inequality when $K=(n+1)\Delta _{n}$ is a copy of the standard simplex in Euclidean $n$ -dimensional space, whose sides are scaled up by a factor of $n+1$. Equivalently, $K=(n+1)\Delta _{n}$ is congruent to the convex hull of the vectors $-\sum _{i=1}^{n}\mathbf {e} _{i}$, and $(n+1)\mathbf {e} _{j}-\sum _{i=1}^{n}\mathbf {e} _{i}$ for all $j=1,\ldots,n$. 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 $(n+1)\Delta _{n}$.