Bounded representation and radial projections of bisectors in normed spaces

TL;DR

This paper introduces the bounded representation of bisectors in finite-dimensional Banach spaces, linking their topology to shadow boundaries and simplifying the analysis of radial projections.

Contribution

It defines a new bounded representation concept for bisectors, connecting topology with shadow boundaries, and proves their topological equivalence in the manifold case.

Findings

Bounded representation visualizes bisectors and shadow boundaries.

Topology of bisectors and their bounded representations are equivalent in the manifold case.

Both are homeomorphic to closed (n-1)-dimensional balls.

Abstract

It is well known that the description of topological and geometric properties of bisectors in normed spaces is a non-trivial subject. In this paper we introduce the concept of bounded representation of bisectors in finite dimensional real Banach spaces. This useful notion combines the concepts of bisector and shadow boundary of the unit ball, both corresponding with the same spatial direction. The bounded representation visualizes the connection between the topology of bisectors and shadow boundaries (Lemma 1) and gives the possibility to simplify and to extend some known results on radial projections of bisectors. Our main result (Theorem 1) says that in the manifold case the topology of the closed bisector and the topology of its bounded representation are the same; they are closed, $(n-1)$-dimensional balls embedded in Euclidean $n$-space in the standard way.