How to prove that every open convex set in $\mathbb{R}^{n}$ is homeomorphic to an open ball? 
If $A,B$ are compacts,, convex with nonempty interiors, prove that they are homeomorphic

My ideia is to find a homeomorphism between $A$ and $B$ and extended to closure, since in this case $\text{int}(\overline{A}) = \text{int}(A)$ (Can I do this?). My question is: prove that $\text{int}(A)$ and $\text{int}(B)$ are homeomorphic? I'm trying to prove that a open convex set in $\mathbb{R}^{n}$ is homeomorphic to an open ball, but I still cannot. Can someone give me a hint?
I didnt look for other questions because I dont want to see the complete solution. I just wanted a hint of how to prove it.

Lastly, is possible to generalize this result for a complete metric space?

Hints:
$(1).\ $ Let $z\in U$, an open and bounded convex set in $\mathbb R^n$. Then, each ray from $z$ intersects $\partial U$ in exactly one point.
$(2).\ $ Set $f(x)=x/\|x\|$, use $(1)$ to show that $f$ restricts to a bijection from $\partial U$ to the sphere.
$(3).$ Extend the inverse of the restricted map in $(2)$ to the ball.
$(4)$. Prove that the map and its inverse are continuous bijections.
