The Vanishing Neighbor aka The Curse of Dimensionality

Quantifying Distance Concentration in $\mathbb{R}^d$
$\mu$ (Mean Dist): -
$\sigma$ (Std Dev): -
Contrast ($\sigma/\mu$): -

In low dimensions, $\sigma/\mu$ is high (wide distribution). In high dimensions, $\sigma/\mu \to 0$. This "crushing" of the distribution into the Red Bins means the relative difference between the nearest and farthest point becomes negligible.

Distance between random points $x, y \sim \mathcal{U}(0,1)^d$