Geometry of the proper asymmetric norm

TL;DR

This paper provides a geometric characterization of a special class of asymmetric norms on real vector spaces, linking their structure to the existence of a specific vector that simplifies the norm's behavior.

Contribution

It introduces a geometric criterion for asymmetric norms where a vector exists making the norm vanish after a certain translation, advancing the understanding of asymmetric norm structures.

Findings

Characterization of asymmetric norms with a specific geometric property

Application to the theory of mutually polar retractions on cones

Provides a new perspective on the structure of asymmetric norms

Abstract

There is given the geometric characterization of an asymmetric norm $q$ on the real vector space $X$, for which exists an $u\in X$ such that $q(x-q(x)u)=0$, for each $x\in X$. The result is used in the theory of mutually polar retractions on cones.