A metric space X has Markov type 2, if for any reversible finite-state Markov chain {Zt} (with Z0 chosen according to the stationary distribution) and any map f from the state space to X, the distance Dt from f(Z0) to f(Zt) satisfies E(D t ) ≤ K tE(D 1) for some K = K(X) < ∞. This notion is due to K. Ball (1992), who showed its importance for the Lipschitz extension problem. However until now, ...

We present two new combinatorial approximation frameworks that are not based on LPduality, or even on linear programming. Instead, they are based on weight manipulation in the spirit of the local ratio technique. We show that the first framework is equivalent to the LP based method of dual fitting and that the second framework is equivalent to an LP-based method which we define and call primal ...

We consider the problem of fitting an n × n distance matrix M by a tree metric T . We give a factor O(min{n, (k log n)}) approximation algorithm for finding the closest ultrametric T under the Lp norm, i.e. T minimizes ‖T, M‖p. Here, k is the number of distinct distances in M . Combined with the results of [1], our algorithms imply the same factor approximation for finding the closest tree metr...

We show that if H is a group of polynomial growth whose growth rate is at least quadratic then the Lp compression of the wreath product Z oH equals max { 1 p , 1 2 } . We also show that the Lp compression of Z oZ equals max { p 2p−1 , 2 3 } and the Lp compression of (Z oZ)0 (the zero section of Z oZ, equipped with the metric induced from Z o Z) equals max { p+1 2p , 3 4 } . The fact that the Hi...

Products of independent identically distributed random stochastic 2 × 2 matrices are known to converge in distribution under a trivial condition. Rates for this convergence are estimated in terms of the minimal Lp-metrics and the Kolmogorov metric and applications to convergence rates of related interval splitting procedures are discussed. AMS subject classifications. Primary: 60F05; secondary:...

We consider the travelling salesman problem (TSP) problem on (the metric completion of) 3-edge-connected cubic graphs. These graphs are interesting because of the connection between their optimal solutions and the subtour elimination LP relaxation. Our main result is an approximation algorithm better than the 3/2-approximation algorithm for TSP in general. © 2004 Elsevier B.V. All rights reserved.

This paper deals with on the continuity of solution to Minkowski problem for Lp torsional measure. For p ? (1, n + 2) (n 2,?), we show that a sequence convex bodies in Rn is convergent Hausdorff metric if measures (associated these bodies) weakly convergent. Moreover, also prove measure continuous respect p.

We study combinatorial modulus on self-similar metric spaces. We give new examples of hyperbolic groups whose boundaries satisfy a combinatorial version of the Loewner property, and prove Cannon’s conjecture for Coxeter groups. We also establish some connections with lp-cohomology.