An Invitation to Tropical Alexandrov Curvature

TL;DR

This paper explores Alexandrov curvature in the tropical projective torus, revealing how triangle types influence curvature and its implications for statistical methods with geometric interpretations.

Contribution

It introduces the study of Alexandrov curvature in tropical geometry, linking triangle combinatorics to curvature properties and providing both proofs and computational insights.

Findings

Positive, negative, zero, and undefined curvatures can coexist.

Triangle types are tightly connected to curvature.

Results are supported by proofs and computational experiments.

Abstract

We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric, which has been useful in various statistical analyses, particularly in phylogenomics. Alexandrov curvature is a generalization of classical Riemannian sectional curvature to more general metric spaces; it is determined by a comparison of triangles in an arbitrary metric space to corresponding triangles in Euclidean space. In the polyhedral setting of tropical geometry, triangles are a combinatorial object, which adds a combinatorial dimension to our analysis. We study the effect that the triangle types have on curvature, and what can be revealed about these types from the curvature. We find that positive, negative, zero, and undefined Alexandrov curvature can exist concurrently in tropical settings and that there is a tight connection between triangle combinatorial type and curvature. Our results are established both by proof and computational experiments, and shed light on the intricate geometry of the tropical projective torus. In this context, we discuss implications for statistical methodologies which admit inherent geometric interpretations. This paper is dedicated to Bernd Sturmfels on the occasion of his 60th birthday.