Example

The set $[0,1]^n$ is a convex, compact subset with nonempty interior of $\mathbf{R}^n$. It follows from proposition 2 of I, p. 123 that it is homeomorphic to the closed euclidean ball. More precisely, for every point $x$ of the interior $]0,1[^n$ and every point $b$ of the open euclidean ball, there exists a homeomorphism of $[0,1]^n$ onto $\mathbf{B}_n$ which maps $x$ to $b$ and induces, by passing to subspaces, a homeomorphism of $]0,1[^n$ onto the open euclidean ball, and also a homeomorphism of the boundary of $[0,1]^n$ onto the sphere $\mathbf{S}_{n-1}$.

It follows that every convex, compact subset with nonempty interior of $\mathbf{R}^n$ is homeomorphic to a cube (loc. cit.).