Let $S = \mathbb{S}^{n-1}$, where this denotes $$\mathbb{S}^{n-1} = \left\{ x \in \mathbb{R}^n: \sum_{i=1}^n x_i^2 = 1 \right\}.$$

Let $x_0 \in S$ be fixed. I want to find the area of the region $$R_n = \left \{ x \in S: ||x - x_0||_2 \leq r_n \right \}.$$ More specifically, I would like to know the ratio of the area of this region over the total area of $S$, i.e., $|R_n|/|S|$. If this is too difficult, a lower bound (for example, an estimate of the order of magnitude in terms of $r_n$) would be great.

I suspect that the this area should be $O(r_n^{n-1}),$ at least when $r_n$ is relatively small. If you think about when $n=2$ and you have a circle, then this region is locally similar to a line segment of length $r_n$. If you have $n=3$ and you're dealing with a spherical shell, then this region is locally a circle of radius $r_n$, and the area should be approximately equal to $\pi r_n^2$. Any references or more precise computation would be great!

  • $\begingroup$ The integral looks doable in general, though I haven't written out the details. The volume does asymptotically approach the volume of an $(n-1)$-ball of radius $r_{n}$ as $r_{n}\to0$. <> Interestingly, when $n = 3$ the area is exactly $\pi r^{2}$ for $0 \leq r \leq 2$ (!!), see for example here. $\endgroup$ Sep 19 at 12:14


You must log in to answer this question.

Browse other questions tagged .