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

TL;DR

This paper establishes linear inequalities that characterize the possible side lengths of polygons in symmetric spaces of noncompact type, linking geometric configurations to algebraic structures via Schubert calculus.

Contribution

It introduces a new system of linear inequalities based on mod 2 Schubert calculus that describes restrictions on polygon side lengths in symmetric spaces.

Findings

Derived homogeneous linear inequalities for polygon side lengths

Connected geometric configurations with algebraic Schubert calculus

Provided a framework for analyzing stability of weighted configurations

Abstract

In a symmetric space of noncompact type X = G/K oriented geodesic segments correspond to points in the Euclidean Weyl chamber. We can hence assign vector-valued side-lengths to segments. Our main result is a system of homogeneous linear inequalities describing the restrictions on the side -lengths of closed polygons. The inequalities are based on the mod 2 Schubert calculus in the real Grassmannians G/P for maximal parabolic subgroups P.