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 can have volume no greater than
(
n
+
1
)
n
/
n
!
:

Vol
(
K
)
≤
(
n
+
1
)
n
n
!
.
Equality is achieved in this inequality when
K
=
(
n
+
1
)
Δ
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
)
Δ
n
is congruent to the convex hull of the vectors
−
∑
i
=
1
n
e
i
, and
n
e
i
. 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
)
Δ
n
.