Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity

Abstract

In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond modulo isometries to vectors in the Euclidean Weyl chamber. We can hence assign vector valued lengths to segments. Our main result is a system of homoge- neous linear inequalities, which we call the generalized triangle inequalities or stability inequalities, describing the restrictions on the vector valued side lengths of oriented polygons. It is based on the mod 2 Schubert calculus in the real Grassmannians G=P for maximal parabolic subgroups P. The side lengths of polygons in Euclidean buildings are stud- ied in the related paper [KLM2]. Applications of the geomet- ric results in both papers to algebraic group theory are given in [KLM3]. © 2009 Applied Probability Trust.