A note on Markov type constants
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 property of being \(\text{CAT}(0)\) among geodesic metric spaces.
scientific article; zbMATH DE number 5551215