Almost empty simplices and Klein polyhedra

Let $\Delta$ be an $n$-simplex in $\mathbb{R}^n$ with integer vertices containing exactly one integer point $a$ distinct from its vertices. We prove that if $a$ is contained in the interior of $\Delta$ or in the relative interior of a facet of $\Delta$, then the volume of $\Delta$ is bounded by a quantity depending only on the dimension $n$; otherwise, the volume of $\Delta$ can be arbitrarily large. This result is applied to derive an asymptotic formula for the average number of vertices of Klein polyhedra. The averaging is taken over the Klein polyhedra of $s$-dimensional integer lattices of fixed determinant $N$, where $N$ is a growing parameter. Such a formula was known only for $s=2$, $3$. If $\Gamma$ is a lattice in $\mathbb{R}^s$ of rank $s$, then its Klein polyhedron is defined as the convex hull of non-zero points of $\Gamma$ contained in a given orthant.