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

Abstract

Classes of an equivalence relation on a module $V$ over a supertropical semiring, called rays, carry the underlying 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 $\mathrm{Ray}(V)$ into convex sets and establish the main tool for analyzing varieties of quasilinear stars in $\mathrm{Ray}(V)$. They provide stratifications of $\mathrm{Ray}(V)$ and, therefore, a finer convex analysis that helps better understand geometric properties.