Extended by Balk Metrics.

Let X be a nonempty set and F(X) be the set of nonempty finite subsets of X. The paper deals with the extended metrics τ : F(X) → R recently introduced by Peter Balk. Balk's metrics and their restriction to the family of sets A with ∣A∣ ⩽ n make possible to consider "distance functions" with n variables and related them quantities. In particular, we study such type generalized diameters diamτn and find conditions under which B → diamτn B is a Balk's metric. We prove the necessary and sufficient conditions under which the restriction τ to the set of A ϵ F(X) with ∣A∣ ⩽ 3 is a symmetric G-metric. An infinitesimal analog for extended by Balk metrics is constructed. [ABSTRACT FROM AUTHOR]