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An invitation to tropical Alexandrov curvature

Carlos Améndola and Anthea Monod

Vol. 14 (2023), No. 2, 181–214
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.

Dedicated to Bernd Sturmfels on the occasion of his 60th birthday.

Keywords
Alexandrov curvature, tropical metric, triangles
Mathematical Subject Classification
Primary: 14T90, 62R01
Milestones
Received: 30 September 2022
Revised: 7 February 2023
Accepted: 9 February 2023
Published: 16 May 2024
Authors
Carlos Améndola
Institute of Mathematics
Technical University Berlin
Germany
Anthea Monod
Department of Mathematics
Imperial College London
United Kingdom