Arnold's seminar
30th November, 1999

Jacques-Olivier Moussafir (Paris-9 Dauphine)

1. An effective algorithm for computing 2-dimensional sails.

A simplicial cone C in R^d is a non-degenerate cone spanned by d vectors.
Consider the set of all integral points lying in C and drop the origin. The
convex hull of the remaining points is called the <<Klein Polyhedron>> K of
C, and V, the border of K, is called the sail of C. If the vectors that span
C belong to Z^d there are finitely many compact faces in V. When d=3 there
is an algorithm that makes it possible to compute the compact faces of
rational sails. This algorithm also makes it possible to make experiments
with 2-dimensional sails, and make a conjecture that generalizes the
Gauss-Kuzmin theorem about coninued fractions.

2. Are <<Klein polyhedra>> polyhedra?

Above I used <<>> because K may not be a closed set and therefore K may not
be a polyhedron. One must add some conditions on C. We slightly generalize
the problem, and consider arbitrary polyhedral sets C in R^d, and K the
convex hull of the integral points they contain. We show with minimal
assumptions on C that K is a closed generalized polyhedron.