Random Metric Spaces and the Universal Urysohn Space.2
نویسنده
چکیده
We introduce a model of the set of all Polish (=separable complete metric) spaces which is the cone R of distance matrices, and consider the geometrical and probabilistic problems connected with this object. We prove that the generic Polish space in the sense of this model is the so called universal Urysohn space which was defined by P.S.Urysohn in the 1920-th. Then we consider the metric spaces with measures (metric triples) and define a complete invariant of its-matrix distribution. We give an intrinsic characterization of matrix distribution and using the ergodic theorem give a new proof of Gromov's reconstruction theorem. A natural construction of a wide class of measures on the cone R is given and for these we show that with probability one the random Polish space is again the Urysohn space. There is a tight link of these questions with metric classification of measurable functions of several arguments and classification of the actions of infinite symmetric group ([4, 8]) Applications to the statistical theory of metric space will follow.
منابع مشابه
Random and Universal Metric Spaces
We introduce a model of the set of all Polish (=separable complete metric) spaces: the cone R of distance matrices, and consider geometric and probabilistic problems connected with this object. The notion of the universal distance matrix is defined and we proved that the set of such matrices is everywhere dense Gδ set in weak topology in the cone R. Universality of distance matrix is the necess...
متن کاملRandom metric spaces and universality
We define the notion of a random metric space and prove that with probability one such a space is isometric to the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the properties of metric (in particular, universal) spaces to the properties of distance matrices. We give examples of other categories in which the randomness and u...
متن کاملA theorem of Hrushovski–Solecki–Vershik applied to uniform and coarse embeddings of the Urysohn metric space
A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph R, the universal Urysohn metric space U, and other related objects. We propose a new proof of the result and show how it can be used to average out uniform and coarse embeddings of U (and its v...
متن کاملThe Urysohn universal metric space and hyperconvexity
In this paper we prove that Urysohn univeral space is hyperconvex. We also examine the Gromov hyperbolicity and hyperconvexity of metric spaces. Using fourpoint property, we give a proof of the fact that hyperconvex hull of a δ-Gromov hyperbolic space is also δ-Gromov hyperbolic.
متن کاملLinearly rigid metric spaces and the embedding problem
We consider the problem of isometric embedding of metric spaces into Banach spaces; and introduce and study the remarkable class of so-called linearly rigid metric spaces: these are the spaces that admit a unique, up to isometry, linearly dense isometric embedding into a Banach space. The first nontrivial example of such a space was given by R. Holmes; he proved that the universal Urysohn space...
متن کامل