doubling

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. The base-2 logarithm of M is called the doubling dimension of X. Euclidean spaces





R


d




{\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. In general, Euclidean space





R


d




{\displaystyle \mathbb {R} ^{d}}

has doubling dimension



Θ
(
d
)


{\displaystyle \Theta (d)}

.

View More On Wikipedia.org