On three early papers by Herbert Busemann

Athanase Papadopoulos (1) , Marc Troyanov (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 Section de Mathematiques [Lausanne]

This paper is a commentary and a reading guide to three papers by Herbert Busemann, Über die Geometrien, in denen die ``Kreise mit unendlichem Radius'' die kürzesten Linien sind." (On the geometries where circles of infinite radius are the shortest lines) (1932), ``Paschsches Axiom und Zweidimensionalität," (Pasch's Axiom and Two--Dimensionality) (1933) and ``Über Räume 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.

Mots clés: Axioms of geometry; Minkowski spaces; Hilbert axioms; Busemann geometry; horospheres.

Domaines: Géométrie métrique [math.MG]; Histoire et perspectives sur les mathématiques [math.HO]; Topologie géométrique [math.GT]

Athanase Papadopoulos, Marc Troyanov. On three early papers by Herbert Busemann. 2016. ⟨hal-01385698⟩