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In this paper we examine the relationship between hyperconvex hulls and metric trees. After providing a linking construction for hyperconvex spaces, we show that the four-point property is inherited by the hyperconvex hull, which leads to the theorem that every complete metric tree is hyperconvex. We also consider some extension theorems for these spaces.
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In this note we prove that on general metric measure spaces the perimeter is equal to the relaxation of the Minkowski content w.r.t. convergence in measure
We give study the Lipschitz continuity of Mobius transformations of a punctured disk onto another punctured disk with respect to the distance ratio metric.
A new similarity invariant metric $v_G$ is introduced. The visual angle metric $v_G$ is defined on a domain $GsubsetneqRn$ whose boundary is not a proper subset of a line. We find sharp bounds for $v_G$ in terms of the hyperbolic metric in the particular case when the domain is either the unit ball $Bn$ or the upper half space $Hn$. We also obtain the sharp Lipschitz constant for a Mobius transformation $f: Grightarrow G$ between domains $G$ and $G$ in $Rn$ with respect to the metrics $v_G$ and $v_{G}$. For instance, in the case $G=G=Bn$ the result is sharp.
The distortion of distances between points under maps is studied. We first prove a Schwarz-type lemma for quasiregular maps of the unit disk involving the visual angle metric. Then we investigate conversely the quasiconformality of a bilipschitz map with respect to the visual angle metric on convex domains. For the unit ball or half space, we prove that a bilipschitz map with respect to the visual angle metric is also bilipschitz with respect to the hyperbolic metric. We also obtain various inequalities relating the visual angle metric to other metrics such as the distance ratio metric and the quasihyperbolic metric.
In this paper we provide several emph{metric universality} results. We exhibit for certain classes $cC$ of metric spaces, families of metric spaces $(M_i, d_i)_{iin I}$ which have the property that a metric space $(X,d_X)$ in $cC$ is coarsely, resp. Lipschitzly, universal for all spaces in $cC$ if the collection of spaces $(M_i,d_i)_{iin I}$ equi-coarsely, respectively equi-Lipschitzly, embeds into $(X,d_X)$. Such families are built as certain Schreier-type metric subsets of $co$. We deduce a metric analog to Bourgains theorem, which generalized Szlenks theorem, and prove that a space which is coarsely universal for all separable reflexive asymptotic-$c_0$ Banach spaces is coarsely universal for all separable metric spaces. One of our coarse universality results is valid under Martins Axiom and the negation of the Continuum Hypothesis. We discuss the strength of the universality statements that can be obtained without these additional set theoretic assumptions. In the second part of the paper, we study universality properties of Kaltons interlacing graphs. In particular, we prove that every finite metric space embeds almost isometrically in some interlacing graph of large enough diameter.
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