Extended by Balk Metrics

Let X be a nonempty set and T(X) be the set of nonempty finite subsets of X. The paper deals with the extended metrics r : (X) R recently introduced by Peter Balk. Balk's metrics and their restriction to the family of sets A with 1A1 n make possible to consider "distance functions" with n variables and related them quantities. In particular, we study such type generalized diameters diam,,, and find conditions under which B -> diam(T)(n) B is a Balk's metric. We prove the necessary and sufficient conditions under which the restriction tau to the set of A is an element of T(X) with vertical bar A vertical bar <= 3 is a symmetric G-metric. An infinitesimal analog for extended by Balk metrics is constructed.