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.
