Back to Search
Start Over
Metric Spaces Admitting Low-distortion Embeddings into All n-dimensional Banach Spaces
- Source :
- Canadian Journal of Mathematics. 68:876-907
- Publication Year :
- 2016
- Publisher :
- Canadian Mathematical Society, 2016.
-
Abstract
- For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional Euclidean spaces, and equilateral spaces. We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that any $n$-point ultrametric can be embedded with uniformly bounded distortion into any Banach space of dimension $\log n$. The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension $n$. This partially answers a question of G. Schechtman.<br />37 pages, 5 figures, some small improvements of presentation
- Subjects :
- Pure mathematics
Computer Science::Information Retrieval
General Mathematics
010102 general mathematics
Dimension (graph theory)
Banach space
Metric Geometry (math.MG)
01 natural sciences
Functional Analysis (math.FA)
Mathematics - Functional Analysis
Distortion (mathematics)
Metric space
Primary: 46B85, Secondary: 05C12, 30L05, 46B15, 52A21
Mathematics - Metric Geometry
0103 physical sciences
Metric (mathematics)
Euclidean geometry
FOS: Mathematics
Uniform boundedness
010307 mathematical physics
0101 mathematics
Ultrametric space
MathematicsofComputing_DISCRETEMATHEMATICS
Mathematics
Subjects
Details
- ISSN :
- 14964279 and 0008414X
- Volume :
- 68
- Database :
- OpenAIRE
- Journal :
- Canadian Journal of Mathematics
- Accession number :
- edsair.doi.dedup.....78028130cb7e56e39424f8cfb5940383