The minimal positive volume of a lattice polyhedron in $\mathbb{R}^{n+1}$ is $\frac{1}{(n+1)!}$. Any lattice polyhedron of volume $\frac{1}{(n+1)!}$ can be identified with the standard simplex $\Delta_{1}$ (see (1)) by an element of $ASL_{n+1}(\mathbb{Z})$.