On three early papers by Herbert Busemann

TL;DR

This paper provides a commentary and reading guide on Herbert Busemann's early foundational works in geometry, highlighting their ideas and techniques that influenced his lifelong research on Minkowski spaces and axiomatic geometry.

Contribution

It offers the first comprehensive analysis and translation of Busemann's early papers, clarifying their significance in the development of geometric foundations and Minkowski space theory.

Findings

Identifies key ideas and techniques from Busemann's early work

Connects early papers to his later research on Minkowski spaces

Provides translations and commentary to facilitate understanding

Abstract

This paper is a commentary and a reading guide to three papers by Herbert Busemann, \"Uber die Geometrien, in denen die "Kreise mit unendlichem Radius" die k\"urzesten Linien sind." (On the geometries where circles of infinite radius are the shortest lines) (1932), "Paschsches Axiom und Zweidimensionalit\"at," (Pasch's Axiom and Two--Dimensionality) (1933) and "\"Uber R\"aume mit konvexen Kugeln und Parallelenaxiom (On spaces with convex spheres and the parallel postulate) (1933). These are the first papers that Busemann wrote on the foundations of geometry and the axiomatic characterization of Minkowski spaces (finite-dimensional normed spaces). The subject of these papers followed Busemann for the rest of his life, and the three papers already contain several ideas and techniques that he developed later on, in his work on the subject which lasted several decades. The three papers were translated into English by Annette A'Campo. These translations, together with the final version of present commentary, will be part of the forthcoming edition of Busemann's Collected Papers edition.