Stratifications of the ray space of a tropical quadratic form by Cauchy-Schwartz functions

TL;DR

This paper introduces a stratification of the ray space in tropical quadratic forms using Cauchy-Schwarz functions, enhancing convex geometric analysis in supertropical algebra.

Contribution

It develops a new stratification method of ray spaces via CS-functions, advancing supertropical trigonometry and convex geometry.

Findings

CS-functions partition ray space into convex sets

Stratifications enable finer convex analysis

Provides new tools for geometric understanding in supertropical settings

Abstract

Classes of an equivalence relation on a module V over a supertropical semiring, called rays, carry the underlaying structure of "supertropical trigonometry" and thereby a version of convex geometry which is compatible with quasilinearity. In this theory the traditional Cauchy-Schwarz inequality is replaced by the CS-ratio which gives rise to special characteristic functions, called CS-functions. These functions partite the ray space Ray(V) into convex sets and establish a main tool for analyzing varieties of quasilinear stars in Ray(V). They provide stratifications of Ray(V) and therefore a finer convex analysis that helps for a better geometric understanding.