Finite Ultrametric Balls

Abstract

The necessary and sufficient conditions under which a given family $\mathcal{F}$ of subsets of finite set $X$ coincides with the family ${\mathbf{B}}_X$ of all balls generated by some ultrametric $d$ on $X$ are found. It is shown that the representing tree of the ultrametric space $({\mathbf{B}}_X,d_H)$ with the Hausdorff distance $d_H$ can be obtained from the representing tree $T_X$ of ultrametric space $(X,d)$ by adding a leaf to every internal vertex of $T_X$.

arXiv comment22 pages, 2 figures

References in corpus (2)

• Products of metric spaces, covering numbers, packing numbers and characterizations of ultrametric spaces
• Weak similarities of finite ultrametric and semimetric spaces