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A metric space $X$ is rigid if the isometry group of $X$ is trivial. The finite ultrametric spaces $X$ with $|X| geq 2$ are not rigid since for every such $X$ there is a self-isometry having exactly $|X|-2$ fixed points. Using the representing trees we characterize the finite ultrametric spaces $X$ for which every self-isometry has at least $|X|-2$ fixed points. Some other extremal properties of such spaces and related graph theoretical characterizations are also obtained.
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Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitati
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 $T_X$.
We study extremal properties of finite ultrametric spaces $X$ and related properties of representing trees $T_X$. The notion of weak similarity for such spaces is introduced and related morphisms of labeled rooted trees are found. It is shown that the finite rooted trees are isomorphic to the rooted trees of nonsingular balls of special finite ultrametric spaces. We also found conditions under which the isomorphism of representing trees $T_X$ and $T_Y$ implies the isometricity of ultrametric spaces $X$ and $Y$.
We study Ramsey-theoretic properties of several natural classes of finite ultrametric spaces, describe the corresponding Urysohn spaces and compute a dynamical invariant attached to their isometry groups.
Let $(X, d)$ be a compact metric space and let $mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I colon mathcal{M}(X) to R$ by $I(mu) = int_X int_X d(x,y) dmu(x) dmu(y)$, and set $M(X) = sup I(mu)$, where $mu$ ranges over the collection of measures in $mathcal{M}(X)$ of total mass 1. The space $(X, d)$ is emph{quasihypermetric} if $I(mu) leq 0$ for all measures $mu$ in $mathcal{M}(X)$ of total mass 0 and is emph{strictly quasihypermetric} if in addition the equality $I(mu) = 0$ holds amongst measures $mu$ of mass 0 only for the zero measure. This paper explores the constant $M(X)$ and other geometric aspects of $X$ in the case when the space $X$ is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are $L^1$-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [Peter Nickolas and Reinhard Wolf, emph{Distance geometry in quasihypermetric spaces. I}, emph{II} and emph{III}].
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