What is the relationship between Curse of Dimensionality and isotropic neighborhoods?

The problem that Hastie, Tibshirani and Friedman are talking about here is that the number of fixed-size neighborhoods goes up exponentially with the dimension. If you're trying to get some intuition for how isotropic neighborhoods are affected by the curse of dimensionality, think about approximating ball-shaped (isotropic) neighborhoods with cube-shaped neighborhoods. Suppose we have an $d$-dimensional unit cube $[0, 1] d$ that we want to divide up into cube-shaped neighborhoods. If I want a neighborhood of side length $\delta 0.1$, in one dimension this requires $10 1 10$ neighborhoods. In two dimensions, this requires $10 2 100$ neighborhoods.