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
Equality is achieved in this inequality when
The conjecture, furthermore, asserts that equality is achieved in the above inequality if and only if K is unimodularly equivalent to