This is an announcement for the paper "Markov convexity and local rigidity of distorted metrics" by Manor Mendel and Assaf Naor.
Abstract: The geometry of discrete tree metrics is studied from the following perspectives: 1. Markov p-convexity, which was shown by Lee, Naor, and Peres to be a property of p-convex Banach space, is shown here to be equivalent to p-convexity of Banach spaces. 2. On the other hand, there exists an example of a metric space which is not Markov p-convex for any finite p, but does not uniformly contain complete binary trees. Note that the previous item implies that Banach spaces contain complete binary trees uniformly if and only if they are not Markov p-convex for any finite p. 3. For every B>4, a metric space X is constructed such that all tree metrics can be embedded in X with distortion at most B, but when large complete binary trees are embedded in X, the distortion tends to B. Therefore the class of finite tree metrics do exhibit a dichotomy in the distortions achievable when embedding them in other metric spaces. This is in contrast to the dichotomy exhibited by the class of finite subsets of L_1, and the class of all finite metric spaces.
Archive classification: math.MG math.FA
Remarks: 10 pages, extended abstract to appear in SoCG '08
%The source file(s), Charlie-tree-socg.bbl: 8435 bytes Charlie-tree-socg.tex: 150202 bytes figs/3path-types.eps: 26109 bytes figs/3path-types.pdf: 13428 bytes figs/d_e-metric.eps: 27160 bytes figs/d_e-metric.pdf: 16009 bytes figs/fork-types.eps: 25081 bytes figs/fork-types.pdf: 12092 bytes figs/lang.eps: 30961 bytes figs/lang.pdf: 14246 bytes figs/mid-lemma1.eps: 18989 bytes figs/mid-lemma1.pdf: 10463 bytes figs/mid-lemma2-c1a.eps: 21267 bytes figs/mid-lemma2-c1a.pdf: 12517 bytes figs/mid-lemma2-c1b.eps: 18219 bytes figs/mid-lemma2-c1b.pdf: 10695 bytes figs/mid-lemma2-c1c.eps: 21626 bytes figs/mid-lemma2-c1c.pdf: 12610 bytes figs/mid-lemma2-c2a.eps: 24273 bytes figs/mid-lemma2-c2a.pdf: 14271 bytes figs/mid-lemma2-c2b.eps: 18207 bytes figs/mid-lemma2-c2b.pdf: 10699 bytes figs/mid-lemma2-c2c.eps: 21237 bytes figs/mid-lemma2-c2c.pdf: 12496 bytes figs/midpoints-new.eps: 17442 bytes figs/midpoints-new.pdf: 9662 bytes figs/tip-contract.eps: 13542 bytes figs/tip-contract.pdf: 6745 bytes figs/type-II-surprise.eps: 17919 bytes figs/type-II-where-w.eps: 11124 bytes sig-alt-full.cls: 56035 bytes, is(are) stored in gzipped form as 0803.1697.tar.gz with size 302kb. The corresponding postcript file has gzipped size 146kb.
Submitted from: mendelma@gmail.com
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