Angles in normed spaces

TL;DR

This paper provides a comprehensive overview of angles in normed spaces, exploring concepts, measures, and orthogonality, including new results and proofs, to fill a gap in the existing literature.

Contribution

It offers the first complete survey of angle concepts in Banach spaces, including new insights and proofs, connecting angles with orthogonality types and applications.

Findings

Analysis of various angle functions and measures in Banach spaces

Introduction of new results and proofs related to angles and orthogonality

Discussion of applications and properties of angular bisectors

Abstract

The concept of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including also further related aspects. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.