Extended by Balk metrics

Let $X$ be a nonempty set and $\mathcal{F}(X)$ be the set of nonempty finite subsets of $X$. The paper deals with the extended metrics $\tau:\mathcal{F}(X)\to\mathbb{R}$ recently introduced by Peter Balk. Balk's metrics and their restriction to the family of sets $A$ with $|A|\leqslant n$ make possible to consider "distance functions" with $n$ variables and related them quantities. In particular, we study such type generalized diameters $\diam_{\tau^n}$ and find conditions under which $B\mapsto\diam_{\tau^n}B$ is a Balk's metric. We prove the necessary and sufficient conditions under which the restriction $\tau$ to the set of $A\in\mathcal{F}(X)$ with $|A|\leqslant 3$ is a symmetric $G$-metric. An infinitesimal analog for extended by Balk metrics is constructed.