Metric Dichotomies

These are notes from talks given at ICMS, Edinburgh, 4/2007 (“Geometry and Algorithms workshop”) and at Bernoulli Center, Lausanne 5/2007 (“Limits of graphs in group theory and computer science”). We survey the following type of dichotomies exhibited by certain classes X of finite metric spaces: For every host space H, either all metrics in X embed almost isometrically in H, or the distortion of embedding some metrics of X in H is unbounded. 1. Problem statement and motivation In these notes we examine dichotomy phenomena exhibited by certain classes X of finite metric spaces. When attempting to embed the metrics in X in any given host spaces H , either all of them embed almost isometrically, or there are some metrics in X which are very poorly embedded in H . To make this statement precise we define the distortion of metric embeddings. Given a mapping between metric spaces f : X → H , define the Lipschitz norm of f to be ‖f‖Lip = supx 6=y dH(f(x), f(y))/dX(x, y). The distortion of injective mapping f is defined as dist(f) = ‖f‖Lip · ‖f‖Lip, where f is defined on f(X). The “least distortion” in which X can be embedded in H is defined as cH(X) = inf{dist(f)| f : X → H}. This is a measure of the faithfulness possible when representing X using a subset of H . We formalize the discussion above as follows: Definition 1 (Qualitative Dichotomy). A class of finite metric spaces X has the qualitative dichotomy property if for any host space H , either • supX∈X cH(X) = 1; or • supX∈X cH(X) = ∞. Remark 1. As defined in Def. 1, the dichotomy is with respect to all metric spaces as hosts. It is possible to extend the definition to be with respect to all sets of metric spaces as hosts. That is, for a set of metric spaces H, define cH(X) = infH∈H cH(X), and replace the use of “cH(X)” in Def. 1 with “cH(X)”. This extension, however, is inconsequential and the two definitions are equivalent. This follows from the proof of Theorem 1.6 in [33], which implies that for any set of metric spaces H, there exists a metric Ĥ, such that for any finite metric space X , cĤ(X) = cH(X). A dichotomy theorem for X can be interpreted as a form of rigidity of X : Small deformations of all the spaces in X is impossible. We will also be interested in stronger dichotomies — of a quantitative nature — in which the unboundedness condition of the distortion is replaced with quantitative estimates on the rate in which it tends to infinity as a function of the size of the 1