Remark

These propositions are not peculiar to Euclidean balls, but can be extended to a whole category of compact neighbourhoods of $o$ in $\mathbf{R}^n$ (see Exercise 12).

The sets $S_{n-1}$ and $B_n$ are evidently invariant under all orthogonal transformations. If $V$ is a $p$-dimensional vector subspace in $\mathbf{R}^n$, there exists an orthogonal transformation which transforms $V$ into a $p$-dimensional coordinate variety; hence $V \cap S_{n-1}$ (resp. $V \cap B_n$) is homeomorphic to $S_{p-1}$ (resp. $B_p$).