Root polytope and partitions

Given a crystallographic reduced root system and an element v of the lattice generated by the roots we study the minimum number |v|, called the length of v, of roots needed to express v as sum of roots. This number is related to the linear functionals presenting the convex hull of the roots; the map v --> |v| turns out to be piecewise quasi-linear with quasi-linearity domains the cones over the facets of this convex hull. In order to show this relation we investigate the integral closure of the monoid generated by the roots in a facet. We study also the positive lenght, i.e. the minimum number of positive roots needed to write an element, and we prove that the two notions of length coincide for type A and C.