2. Embeddings
-
Deterministic proofs of bi-Lipschitz embedding theorems
Assouad’s embedding theorem asserts that to each doubling metric space (X,d) and each \epsilon \in (0,1), there corresponds n so that the snowflake metric space (X,d^\epsilon) admits a bi-Lipschitz embedding into {\mathbb R}^n. Naor–Neiman proved that n can be chosen independent of the snowflake parameter \epsilon, for \epsilon near one (say, \epsilon>\tfrac12). Their proof is probabilistic and nonconstructive. Can one give an explicit construction of such embedding?Problem 2.1.
[Mario Bonk] Give a deterministic proof of Naor–Neiman’s improved Assouad embedding theorem.-
Remark. Solved by Guy David and Marie Snipes in "A Non-Probabilistic Proof of the Assouad Embedding Theorem with Bounds on the Dimension." Analysis and Geometry in Metric Spaces 1 (2013): 36-41. <http://eudml.org/doc/267210>
-
-
Curvature conditions and bi-Lipschitz embeddings
Problem 2.3.
[Leonid Kovalev] Let X be a doubling metric space. Can one formulate a curvature-type condition which, if satisfied by X, ensures that X admits a bi-Lipschitz embedding into some finite-dimensional Euclidean space? -
Bi-Lipschitz embeddings: Hilbert spaces vs. finite-dimensional Euclidean spaces
Problem 2.4.
Suppose that X is a doubling space which admits a bi-Lipschitz embedding into Hilbert space. Does X necessarily admit a bi-Lipschitz embedding into some finite-dimensional Euclidean space? -
Snowflake embeddings of the Heisenberg group into low-dimensional Euclidean spaces
Let d_{cc} denote the Carnot-Carathéodory metric.Problem 2.4.
[Enrico Le Donne] Does there exist \epsilon<1 so that the first Heisenberg group {\mathbb H}^1 equipped with the metric d_{cc}^\epsilon bi-Lipschitz embeds into {\mathbb R}^5?-
Remark. Instead one could use the Korányi metric d_0(p,q) = ||p^{-1}*q||_0, where ||(z,t)||_0 = (|z|^4+t^2)^{1/4}.
-
Cite this as: AimPL: Mapping theory in metric spaces, available at http://aimpl.org/mappingmetric.