A Union of Euclidean Metric Spaces is Euclidean
A Union of Euclidean Metric Spaces is Euclidean, Discrete Analysis 2016:14, 15pp.
A major theme in metric geometry concerns conditions under which it is possible to embed one metric space into another with small distortion. More precisely, if M1 and M2 are metric spaces, one would like to find a function f:M1→M2 and constants c and C such that for every x,y∈M1 the inequalities cd(x,y)≤d(f(x),f(y))≤Cd(x,y) hold, with the ratio C/c being as small as possible. For example, the famous Johnson-Lindenstrauss lemma states that if a1,…,an are points in a Euclidean space and ϵ>0, then there are points b1,…,bn that live in a Euclidean space of dimension at most 8logn/ϵ2 such that
for every i,j. That is, a finite subset A of a Euclidean space can be embedded with small distortion into a Euclidean space with dimension logarithmic in the size of A.
A rather basic question one can ask about embeddings of small distortion is the following. Let A and B be two subspaces of a metric space X and suppose that each can be embedded into Euclidean space with bounded distortion. Can their union A∪B also be embedded with bounded distortion? One might think that this question would have either a simple proof or a simple counterexample, but surprisingly it was a question asked by Assaf Naor that had been open for some time. It is solved in this paper. If Z is a metric space, write DZ for the smallest distortion with which one can embed Z into Euclidean space – that is, the smallest possible ratio C/c in the discussion above. The main result of the paper is that
In particular, if DA and DB are finite, then so is DA∪B.
This result leaves open several further interesting questions, mentioned at the end of the paper. In particular, the ingenious proof uses a specifically Hilbertian tool, the Kirszbraun extension theorem, so it does not solve the problem for embeddings into other spaces, such as ℓ1. It would also be interesting to know what the best possible dependence is of DA∪B on DA and DB. It is likely that this paper will stimulate many further interesting developments in metric geometry.