New intrinsic four point properties which characterize hyperbolic and euclidean spaces

Abstract

Many characterizations of euclidean spaces (real inner product spaces) among metric spaces have been based on euclidean four-point embeddability properties. Related "intrinsic" four point properties have also been used to characterize euclidean or hyperbolic spaces among a suitable class of metric spaces. The present paper provides new characterizations of euclidean or hyperbolic spaces based on intrinsic four point properties which are related to known euclidean four point embedding properties, and in addition provides a new euclidean four-point embedding property which characterizes euclidean spaces among metric spaces. Application of this property in normed linear spaces provides an apparently new characterization of inner product spaces among normed linear spaces.