  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.
