Question

What's a good reference e.g. textbook for the fact that the open n-ball and the open n-simplex are diffeomorphic?

Answer 1

If the compact simplex is

$$\Delta_n = \{ (x_0,\cdots,x_n) : x_i \geq 0, x_0+x_1+\cdots+x_n=1\} \subset \mathbb R^{n+1}$$

then consider this function $f : \Delta \to \mathbb R \cup \{\infty\}$ defined by

$$f(x_0,\cdots,x_n) = \frac{1}{x_0} + \cdots + \frac{1}{x_n}$$

This is a proper Morse function on the interior of $\Delta_n$, and there's only the one critical point at $(\frac{1}{n+1},\cdots,\frac{1}{n+1})$, so standard theorems in Morse theory give you a diffeo to the open ball.

I imagine this is simple enough that you could solve the corresponding ODEs explicitly and write the diffeo out in a closed-form but I haven't put in the work.

Answer 2

I'd recommend taking a look at the book by Brocker and Janich, which discusses diffeomorphisms between star-shaped domains by defining a flow along rays from the star point. This might be an exercise in the book, rather than a theorem. (I think I have the book on my desk at school, and I'll try to give a more precise reference tomorrow.)