Convex compact set with non-empty interior is homeomorphic to closed unit ball

Question

I'm trying to prove that any compact convex subset of $\mathbb{R}^n$ with non-empty interior is homeomorphic to the closed unit ball. Any suggestions on how to prove it? So far I'm thinking that I could translate the subset so that the origin is in the interior of the subset, and then draw rays from the origin to the boudary of the subset and contract those rays until they fill the closed unit ball, but I'm not sure if it will work. Any help is really appreciated.

Answer 1

Suppose (without loss of generality) that $0 \in \mathring{K}$ your compact convex set (you can translate, it is an homeo). Then we define a norm on $\mathbb{R}^n$ given by $N(x) = \frac{\|x\|_2}{\|x_K \|_2}$ for $x\neq 0$, where $x_K$ is the unique element that maximizes the function: $$y \in \mathbb{R}_+ x \cap K \mapsto \|y\|_2$$

It is unique for obvious reasons (this is hemeomorphic to a compact non-empty interval, and on this interval, this function is strictly increasing) and lies in $K$.

---

NB:$x_K$ is the intersection of $\mathbb{R}_{\geq 0}x$ with $\partial K$.

---

then your homeomorphism is given by $$x \in K \mapsto \frac{N(x)}{\|x\|_2} \cdot x$$ if $x \neq 0$ and $0$ otherwise

It is quite obvious that it is bijective (restrict it to lines and you get the result), but the bicontinuity can be a hassle. Typically the hard part should be the bicontinuity at $0$.

Hint for the bicontinuity: use norm equivalence on well chosen subsets of $K$ a recursion may be needed