To capture a fixed fraction f of data in p dimensions, you need r = f1/p of each axis. In 10D, capturing 10% of data requires reaching 79% of each axis.
0.10
2D Square Analogy
Circle radius = r(p) at selected f.
As p grows, "local" fills the square.
Table: r(p) at f = 0.10
p
r = f^(1/p)
Formula: r = f1/p
f = fraction of data
p = dimension
r = side-length of hypercube
needed (range 0–1)