Hyperbolic and euclidean distance functions

TitleHyperbolic and euclidean distance functions
Publication TypeJournal Article
Year of Publication1997
AuthorsBenz W
JournalAnnuaire de l’Université de Sofia “St. Kliment Ohridski”. Faculté de Mathématiques et Informatique
Volume89
IssueLivre 1 - Mathématiques et Mecanique
Pagination59-67
ISSN0205-0808
Keywordshyperbolic distance, invariance of distance functions under special motions
Abstract

This is a functional equations approach to the non-negative functions $h(x,y)$ and $e(x,y)$ as defined in formulas (1) and (2). Moreover, all distance functions of $\mathbb{R}_{n}$ are characterized, which are invariant under linear and orthogonal mappings (see Theorem 1), and, especially, all functions of this type are determined, which satisfy in addition ($D_{2}$) (see Theorem 2). Here ($D_{2}$) asks for the invariance under euclidean or hyperbolic translations of the $x_{1}$-axis. Finally, additivity on the $x_{1}$-axis is considered, leading to the distance functions $h$ and $e$ up to non-negative factors (see Theorem 3).

1991/95 MSC

39B40, 51M10, 51K05

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