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

TL;DR

This paper explores the structure of ray spaces in supertropical quadratic forms using Cauchy-Schwarz functions, introducing convex partitions and geometric concepts tailored to supertropical algebra.

Contribution

It develops a convex geometric framework in supertropical ray spaces utilizing CS-functions, enabling finer analysis of quasilinear structures and partitions.

Findings

CS-functions induce convex partitions of ray space

The framework supports supertropical trigonometry and convex analysis

Provides tools for understanding quasilinear stars in ray space

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 which emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space Ray(V). In particular, these functions induce a partition of Ray(V) into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars.