Doubling space

In mathematics, a metric space X with metric d is said to be doubling if there is some doubling constant M > 0 such that for any x ∈ X and r > 0, it is possible to cover the ball B(x, r) = {y | d(x, y) < r} with the union of at most M balls of radius ⁠r/2⁠.

[1] The base-2 logarithm of M is called the doubling dimension of X.

Doubling metric spaces, on the other hand, would seem like they have more of a chance, since the doubling condition says, in a way, that the metric space is not infinite dimensional.

The Heisenberg group with its Carnot-Caratheodory metric is an example of a doubling metric space which cannot be embedded in any Euclidean space.

[5] Assouad's Theorem states that, for a M-doubling metric space X, if we give it the metric d(x, y)ε for some 0 < ε < 1, then there is a L-bi-Lipschitz map

One example on the real line is the weak limit of the following sequence of measures:[9] One can construct another singular doubling measure μ on the interval [0, 1] as follows: for each k ≥ 0, partition the unit interval [0,1] into 3k intervals of length 3−k.

[10] The definition of a doubling measure may seem arbitrary, or purely of geometric interest.

However, many results from classical harmonic analysis and computational geometry extend to the setting of metric spaces with doubling measures.