Klein polyhedra and relative minima of lattices

We prove that in $\mathbb R^3$, the relative minima of almost any lattice belong to the surface of the corresponding Klein polyhedron. We also prove, for almost any lattice in $\mathbb R^3$, that the set of relative minima with nonnegative coordinates coincides with the union of the set of extremal points of the Klein polyhedron and a set of special points belonging to the triangular faces of the Klein polyhedron.