ARNOLD POLYHEDRON AND SAIL OF A SIMPLICIA L CONE - ANALOGS OF LAGRANGE THEOREM

The sail V of a simplicial cone C of the euclidean space of dimension d is the boundary of the convex hull of the integer points included in C. We show that the cone C is algebraic in some sense if and only if the faces (or the generating rays) of C admit a periodic approximation by a path of simplexes (or of extremal points) of V. If C is split on a totally real field of degree d, the group of integral matrices stab ilizing V admits a fundamental set which is the union of a finite numb er of simplexes of V.