Strongly non-embeddable metric spaces
Definition. Let \(p\geq 0\). A metric space \((X,d)\) is said to have \textit{generalized roundness} \(p\) if for all \(n\in\mathbb{N}\) and all \(a_1,\dots,a_n, b_1, \dots, b_n\in X\) the inequality NEWLINE\[NEWLINE\sum_{1\leq k<\ell\leq n}\left(d(a_k,a_\ell)^p+d(b_k,b_\ell)^p\right)\leq \sum_{1\leq j,i\leq n}d(a_j,b_i)^p NEWLINE\]NEWLINE holds. \textit{P. Enflo} [Ark. Mat. 8, 107--109 (1970; Zbl 0196.14003)] constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. \textit{A. N. Dranishnikov, G. Gong, V. Lafforgue} and \textit{G. Yu} [Can. Math. Bull. 45, No. 1, 60--70 (2002; Zbl 1044.46020)] modified Enflo's example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. The authors of this paper combine these two examples and get a locally finite metric space which may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. It is important to mention that (1) by a uniform embedding the authors mean a uniformly continuous map \(f\) having uniformly continuous inverse \(f^{-1}\) defined on the image of \(f\); (2) in their definition of a locally finite metric space the authors do not require that \(\inf_{x\neq y}d(x,y)>0\).NEWLINENEWLINEThe authors make some related observations on Cayley graphs of infinite abelian groups and on Lipschitz injections.
scientific article; zbMATH DE number 6023691
scientific article