Note on bi-Lipschitz embeddings into normed spaces
نویسنده
چکیده
Let (X, d), (Y, ρ) be metric spaces and f : X → Y an injective mapping. We put ‖f‖Lip = sup{ρ(f(x), f(y))/d(x, y); x, y ∈ X, x 6= y}, and dist(f) = ‖f‖Lip.‖f ‖Lip (the distortion of the mapping f). We investigate the minimum dimension N such that every n-point metric space can be embedded into the space lN ∞ with a prescribed distortion D. We obtain that this is possible for N ≥ C(logn)2n3/D, where C is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into lp are obtained by a similar method.
منابع مشابه
Quantitative Bi-Lipschitz embeddings of bounded curvature manifolds and orbifolds
We construct bi-Lipschitz embeddings into Euclidean space for bounded diameter subsets of manifolds and orbifolds of bounded curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. We also construct global bi-Lipschitz embeddings for spaces of the form Rn/Γ , where Γ is a discrete group acting properly discontinuously and by isometries o...
متن کاملAlmost Bi-lipschitz Embeddings and Almost Homogeneous Sets
This paper is concerned with embeddings of homogeneous spaces into Euclidean spaces. We show that any homogeneous metric space can be embedded into a Hilbert space using an almost bi-Lipschitz mapping (biLipschitz to within logarithmic corrections). The image of this set is no longer homogeneous, but ‘almost homogeneous’. We therefore study the problem of embedding an almost homogeneous subset ...
متن کاملLow-distortion embeddings of infinite metric spaces into the real line
We present a proof of a Ramsey-type theorem for infinite metric spaces due to Matoušek. Then we show that for every K > 1 every uncountable Polish space has a perfect subset that K-bi-Lipschitz embeds into the real line. Finally we study decompositions of infinite separable metric spaces into subsets that, for some K > 1, K-bi-Lipschitz embed into the real line.
متن کاملAmenability, Locally Finite Spaces, and Bi-lipschitz Embeddings
We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, clearly extends amenability to any locally finite space. Therefore, we start making a comparison between this property and other notions of amenability for locally finite metric spaces that have been...
متن کاملLeast-Distortion Euclidean Embeddings of Graphs: Products of Cycles and Expanders
Embeddings of finite metric spaces into Euclidean space have been studied in several contexts: The local theory of Banach spaces, the design of approximation algorithms, and graph theory. The emphasis is usually on embeddings with the least possible distortion. That is, one seeks an embedding that minimizes the bi-Lipschitz constant of the mapping. This question has also been asked for embeddin...
متن کامل