A note on Markov type constants
type: Item
instance of: scholarly article
title: A note on Markov type constants
zbMATH Open document ID: 1187.53079
author: Mikaël Pichot
author: Shin-Ichi Ohta
published in: Archiv der Mathematik
publication date: 2009-05-06T00:00:00Z
review text: The classical notions of Rademacher type and cotype have been proven to be of fundamental significance within the local theory of Banach spaces. Generalizations of these concepts to arbitrary metric spaces were introduced in various ways and turned out to be extremely useful w.r.t. understanding the extension problem of Lipschitz maps on the one hand, and the theory of bi-Lipschitz embeddings of finite metric spaces or graphs into \(l_p\)-spaces, on the other and. In the paper under review, the authors focus on the following notion of type, called Markov type: A metric space \((X,d)\) is said to have \textit{Markov type 2} if there is a constant \(K \geq 1\) such that given any \(N \in \mathbb{N}\) and \((x_i) \in X^N\), and any \((\pi_i)_{i=1}^N\) and \(A=(a_{ij})_{i,j=1}^N\) satisfying \[ 0\leq \pi_i, \;\;\; a_{ij}\leq 1, \;\;\; \sum_{i=1}^N \, \pi_i \, = \, 1, \;\;\; \sum_{j=1}^N \, a_{ij} \, = \, 1, \;\;\; \pi_i a_{ij} \, = \pi_j a_{ji}, \] for all \(i,j=1,\dots,N\), one has \(\mathcal{E}(l)\leq K^2l\mathcal{E}(1)\) for all \(l\in \mathbb{N}\), where \(\mathcal{E}(l):=\sum_{i,j=1}^N \, \pi_i a_{ij}^{(l)}d(x_i,x_j)^2\). The least such constant \(K\) is denoted by \(M_2(X)\) and is called the \textit{Markov type \(2\) constant} of \(X\). Using K. T. Sturm's characterization of Alexandrov spaces, the authors prove that, if a geodesic metric space \(X\) has Markov type \(2\) with constant \(M_2(X)\), then \(X\) is an Alexandrov space of non-negative curvature. The same technique provides a lower bound of the Markov type 2 constant of a space containing a tripod or a branching point. Note that in a subsequent paper Ohta proves that Alexandrov spaces of non-negative curvature have Markov type \(2\) with Markov type \(2\) constant \(M_2(X)\leq 1+\sqrt{2}\). Note further that the authors point out that due to a recent characterization of \(\text{CAT}(0)\)-spaces by Berg and Nikolaev, Enflo type \(2\) with corresponding Enflo type \(2\) constant \(K=1\) characterizes the…
reviewed by: Thomas Foertsch
Mathematics Subject Classification ID: 53C70
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 60J10
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 5551215
zbMATH Keywords: Markov type
zbMATH Keywords: Alexandrov space
zbMATH Keywords: tripod
zbMATH Keywords: geodesic metric space
zbMATH Keywords: Enflo type
MaRDI profile type: Publication
full work available at URL: s00013-008-2672-2
OpenAlex ID: W2071784063
DOI: 10.1007/S00013-008-2672-2
description: scientific article; zbMATH DE number 5551215