  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.
