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**Actes de la table ronde de Géométrie Différentielle en l'honneur de Marcel Berger**

Arthur L. Besse (éditeur)

Séminaires et Congrès **1** (1996), xviii+642 pages

**Cut loci and distance spheres on Alexandrov surfaces**

Katsuhiro Shiohama - Minoru Tanaka

Séminaires et Congrès **1** (1996), 531-559

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**Résumé :**

L'objet de cet article est d'étudier la structure des sphères de distance et du cut locus *C*(*K*) d'un ensemble compact.

**Abstract:**

The purpose of the present paper is to investigate the structure of distance spheres and cut locus *C*(*K*) to a compact set *K* of a complete Alexandrov surface *X* with curvature bounded below. The structure of distance spheres around *K* is almost the same as that of the smooth case. However *C*(*K*) carries different structure from the smooth case. As is seen in examples of Alexandrov surfaces, it is proved that the set of all end points *C*_{e}(*K*) of *C*(*K*) is not necessarily countable and may possibly be a fractal set and have an infinite length. It is proved that all the critical values of the distance function to *K* is closed and of Lebesgue measure zero. This is obtained by proving a generalized Sard theorem for one-valuable continuous functions.

Our method applies to the cut locus to a point at infinity of a noncompact *X* and to Busemann functions on it. Here the structure of all co-points of asymptotic rays in the sense of Busemann is investigated. This has not been studied in the smooth case.

**Class. math. :** 53C20