Cauchy–Schwarz functions and convex partitions in the ray space of a supertropical quadratic form

Abstract

Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a ‘supertropical trigonometry’ and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space . In particular, these functions induce a partition of into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars.