On Bisectors in Normed Spaces

TL;DR

This paper characterizes the geometric shape of bisectors in two-dimensional normed spaces, revealing how the shape depends on the points' positions relative to the unit circle, and describes their structure as cones and curves.

Contribution

It provides a complete description of bisectors in 2D normed spaces, detailing their structure based on point positions and extending understanding of geometric properties in these spaces.

Findings

Bisectors can be composed of cones and curves.

The shape depends on the points' relation to the unit circle.

Bisectors of non-strict pairs resemble those of strict pairs.

Abstract

In this note, we completely describe the shape of the bisector of two given points in a two-dimensional normed vector space. More precisely, we show that, depending on the position of two given points with respect to the shape of the unit circle, the following holds: the bisector of a non-strict pair of points consists of two cones and a curve, which has properties similar to those of bisectors of strict pairs of points.