Open Unit Ball diffeomorphic to the Open Unit Cube How can I show that 
the open unit cube $(-1,1)^n \subset \mathbb{R}^n$ and
the open unit ball $B = \{x \in \mathbb{R}^n  \mid  \|x\| < 1\}$
are diffeomorphic?
I know that one can proof this by showing that those two sets are diffeomorphic to the whole space $\mathbb{R}^n$. But is there a direct way (not over the $\mathbb{R}^n$ ) to proof this? Thus, is there a smooth, differentiable bijection between the two sets with a differentiable inverse?
 A: $\newcommand{\Reals}{\mathbf{R}}$Let $f:(-1, 1) \to \Reals$ be your favorite smooth, increasing bijection, such as
$$
f(x) = \tanh^{-1} x
  \quad\text{or}\quad f(x) = \frac{x}{1 - x^{2}}
  \quad\text{or}\quad f(x) = \tan \tfrac{\pi}{2} x.
$$
The mapping
$$
\phi(x_{1}, \dots, x_{n})
  = \frac{\bigl(f(x_{1}), \dots, f(x_{n})\bigr)}{\sqrt{1 + f(x_{1})^{2} + \dots + f(x_{n})^2}}
$$
is obviously a diffeomorphism from the open cube $C$ to the open unit ball $B$.
Oh, that fact isn't obvious...?
But look: If we define $\psi_{1}:C \to \Reals^{n}$ and $\psi_{2}:\Reals^{n} \to B$ by
$$
\psi_{1}(x_{1}, \dots, x_{n}) = \bigl(f(x_{1}), \dots, f(x_{n})\bigr)
  \quad\text{and}\quad
  \psi_{2}(y) = \frac{y}{\sqrt{1 + \|y\|^{2}}},
$$
it's obvious $\psi_{1}$ and $\psi_{2}$ are diffeomorphisms, and $\phi = \psi_{2} \circ \psi_{1}$.

The deeper point of this imaginary dialogue is, constructing a diffeomorphism from $C$ to $B$ by "going through $\Reals^{n}$" is more elegant than trying to find a "direct" mapping: In fancy terms, using $\Reals^{n}$ as "intermediary" exploits the common one-point compactification of $C$ and $B$.
By contrast, a "direct" mapping (presumably, a diffeomorphism from $B$ to $C$ that extends to the respective boundaries) cannot be smooth at the boundary of $B$. One finds oneself visiting the land of unnecessary vexation.(1)


*

*One natural approach, along the lines of Giuseppe Negro's comment, would be to map the sphere of radius $p$ in the $2$-norm to a sphere of suitable "radius" in the $\frac{1}{1 - p}$-norm, so the images of spherical shells become "more nearly squared off" as the radius increases. This looks doable, though I haven't tried to carry out the details.

