The statistical properties of 3D Klein polyhedra

Let $\Gamma$ be a rank-$s$ lattice in $\mathbb R^s$. The convex hulls of the nonzero lattice points lying in orthants are called the Klein polyhedra of $\Gamma$. This construction was introduced by Klein in 1895, in connection with generalizing the classical continued-fraction algorithm to the multidimensional case. Arnold stated a number of problems on the statistical and geometric properties of Klein polyhedra. In two dimensions the corresponding results follow from the theory of continued fractions. An asymptotic formula for the mean value of the $f$-vectors (the numbers of facets, edges and vertices) of 3D Klein polyhedra is derived. This mean value is taken over the Klein polyhedra of integer 3D lattices with determinants in $[1,R]$, where $R$ is an increasing parameter.
Bibliography: 27 titles.