We prove several results of the following type: given finite dimensional normed space V there exists another space X with log dimX = O(log dimV ) and such that every subspace (or quotient) of X, whose dimension is not “too small,” contains a further subspace isometric to V . This sheds new light on the structure of such large subspaces or quotients (resp., large sections or projections of conve...

Let ‖ · ‖ be a norm on Rn. Averaging ‖(ε1x1, · · · , εnxn)‖ over all the 2n choices of −→ε = (ε1, · · · , εn) ∈ {−1,+1}n, we obtain an expression ‖|x‖| which is an unconditional norm on Rn. Bourgain, Lindenstrauss and Milman [3] showed that, for a certain (large) constant η > 1, one may average over ηn (random) choices of −→ ε and obtain a norm that is isomorphic to ‖| · ‖|. We show that this i...

We prove that if a metric probability space with a usual concentration property embeds into a Banach space X, then X has a proportional Euclidean subspace. In particular, this yields a new characterization of weak cotype 2. We also find optimal lower estimates on embeddings spaces with concentration properties (i.e. uniformly convex spaces) into l ∞, thus providing an ”isomorphic” extension to ...

In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso (U) of isometries of Urysohn’s universal complete separable metric space U, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The ...

[59] Milman Parry’s work attempted to demonstrate the traditional character of Homeric poetry. Despite a later shift in focus from the concept of “traditional” to “oral” (Finnegan 1991: 120), with at least the implication of a relatively open system as opposed to a relatively closed one, Parry’s early emphasis on tradition was so strong that his theory tended to be read as one discounting, if n...

In this note, we prove several of the main results of compressive sensing from the spherical section property. 1 Compressive sensing In the past four years, there has been extensive activity on compressive sensing. Recently, Kashin and Temlyakov [7] and Zhang [8] have developed simplified proofs of some of the main theorems of compressive sensing using the spherical section property. This notes...

Rights: © 1993 American Physical Society (APS). This is the accepted version of the following article: Seitsonen, A. P. & Puska, M. J. & Alatalo, M. & Nieminen, Risto M. & Milman, V. & Payne, M. C. 1993. Crystals from metallic clusters: A first-principles calculation. Physical Review B. Volume 48, Issue 3. 1981-1983. ISSN 1550-235X (electronic). DOI: 10.1103/physrevb.48.1981, which has been pub...

We prove that the isometry group Iso (U) of the universal Urysohn metric space U equipped with the natural Polish topology is a Lévy group in the sense of Gromov and Milman, that is, admits an approximating chain of compact (in fact, finite) subgroups, exhibiting the phenomenon of concentration of measure. This strengthens an earlier result by Vershik stating that Iso (U) has a dense locally fi...

We show that if the Banach-Mazur distance between an n-dimensional normed space X and l∞ is at most 3/2, then there exist n+ 1 equidistant points in X. By a well-known result of Alon and Milman, this implies that an arbitrary n-dimensional normed space admits at least e √ log n equidistant points, where c > 0 is an absolute constant. We also show that there exist n equidistant points in spaces ...