The generalized triangle inequalities in thick Euclidean buildings of rank 2

TL;DR

This paper characterizes the possible side lengths of polygons in rank 2 thick Euclidean buildings using a finite set of linear inequalities derived from the building's combinatorial structure.

Contribution

It introduces the generalized triangle inequalities that describe side length configurations in rank 2 Euclidean buildings, linking geometric and combinatorial properties.

Findings

Set of possible side lengths characterized by linear inequalities

Inequalities depend on the combinatorics of the spherical Coxeter complex

Provides a finite, explicit description of side length configurations

Abstract

We describe the set of possible vector valued side lengths of n-gons in thick Euclidean buildings of rank 2. This set is determined by a finite set of homogeneous linear inequalities, which we call the generalized triangle inequalities. These inequalities are given in terms of the combinatorics of the spherical Coxeter complex associated to the Euclidean building.