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.^[2] Euclidean spaces ${\displaystyle \mathbb {R} ^{d}}$ equipped with the usual Euclidean metric are examples of doubling spaces where the doubling constant M depends on the dimension d. For example, in one dimension, M = 3; and in two dimensions, M = 7.^[3] In general, Euclidean space ${\displaystyle \mathbb {R} ^{d}}$ has doubling dimension ${\displaystyle \Theta (d)}$.^[2][4]