High-dimensional balls behave very differently from ordinary three-dimensional ones. For a unit $n$-ball, the fraction of volume contained inside radius $r<1$ is

$$\frac{V_n(r)}{V_n(1)}=r^n.$$

A thin outer shell contains most of the volume

Take an outer shell of thickness $\varepsilon$. The fraction of volume inside the smaller radius $1-\varepsilon$ is

$$(1-\varepsilon)^n.$$

As $n$ grows, this quantity approaches zero rapidly. Therefore the fraction contained in the outer shell,

$$1-(1-\varepsilon)^n,$$

approaches one.

Most of the volume of a high-dimensional ball lies close to its boundary.

Unit ball versus unit cube

The unit $n$-ball has volume

$$\frac{\pi^{n/2}}{\Gamma(n/2+1)},$$

while a side-$2$ hypercube has volume $2^n$. Their ratio eventually becomes extremely small as dimension increases.

This does not mean the ball is somehow defective. It means the cube contains increasingly vast corner regions far from its center.

Curse of dimensionality

Many numerical and statistical methods become difficult in high dimensions because space grows so quickly that finite data become sparse. Covering a region at fixed resolution can require a number of samples that grows exponentially with dimension.

This geometric effect is one reason high-dimensional modeling often relies on additional structure such as smoothness, sparsity or lower-dimensional representations.