Convex Hulls of Integral Points

The convex hull of all integral points contained in a compact polyhedron C is obviously a compact polyhedron. If C is not compact, then the convex hull K of its integral points need not be a closed set. However, under some natural assumptions, K is a closed set and a generalized polyhedron. Bibliography: 11 titles.     

Sails and Hilbert bases

A Klein polyhedron is the convex hull of the nonzero integral points of a simplicial coneC⊂ ℝⁿ. There are relationships between these polyhedra and the Hilbert bases of monoids of integral points contained in a simplicial cone. In the two-dimensional case, the set of integral points lying on the boundary of a Klein polyhedron contains a Hilbert base of the corresponding monoid. In general, this is not the case if the dimension is greater than or equal to three (e.g., [2]). However, in the three-dimensional case, we give a characterization of the polyhedra that still have this property. We give an example of such a sail and show that our criterion does not hold if the dimension is four. CEREMADE, University Paris 9. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 34, No. 2, pp. 43–49, April–June, 2000. Translated by J.-O. Moussafir     

On computing the entropy of braids

We consider the problem of computing the entropy of a braid. We recall its definition and for each braid construct a sequence of real numbers whose limit is the braid’s entropy. We state one conjecture on the convergence speed and two conjectures on braids that have high entropy but are written with few letters.