  It is well known that the number of tilting modules over a path algebra of
type A_n coincides with the Catalan number C(n). Moreover, the number of
support tilting modules of type A_n is the Catalan number C(n+1). We show that
the convex hull of all roots of a root system of type A_n is a polytope with
integral volume (n + 1)C(n+1). Moreover, we associate to the set of tilting
modules and to the set of support tilting modules certain polytopes and show
that their volumes coincide with the number of those modules, respectively.
Finally, we show that these polytopes can be defined just using the root system
and relate their volumes, so that we can derive the above results in a new way.
