Angles and a Classification of Normed Spaces

TL;DR

This paper introduces a generalized concept of angles in normed spaces based on the shape of the unit ball, leading to a classification of these spaces and a characterization of inner product spaces, including non-convex cases.

Contribution

It proposes a novel way to define angles in arbitrary normed spaces and uses this to classify spaces and identify inner product spaces.

Findings

Defined angles for all real numbers using unit ball shape

Classified normed spaces based on these angles

Characterized inner product spaces among normed spaces

Abstract

We suggest a concept of generalized `angles' in arbitrary real normed vector spaces. We give for each real number a definition of an `angle' by means of the shape of the unit ball. They all yield the well known Euclidean angle in the special case of real inner product spaces. With these different angles we achieve a classification of normed spaces, and we obtain a characterization of inner product spaces. Finally we consider this construction also for a generalization of normed spaces, i.e. for spaces which may have a non-convex unit ball.