contractible open sets

Question

If $U\subset\mathbb{R}^n$ is an open and contractible subset such that there is a continuous function $f\colon U\to\mathbb{R}$ with only one minimum and the level curves of $f$ are connected by paths, then is $U$ homeomorphic to $\mathbb{R}^n$?

Answer 1

$1^{st}$ hint: We may assume that the minimum lies at the origin and that the value at this point is zero.

$2^{nd}$ hint: The map $\mathbb R^n\to\mathbb R$, $x\mapsto ||x||$ has the required property. It is likely that the homeomorphism should send level curves to circles around the origin.

$3^{rd}$ hint: A candidate for the homeomorphism is the map which sends $x\mapsto f(x)\frac{x}{||x||}$ (and $0\mapsto 0$).

Edit: It is relatively easy to see that my propsed map doesn't work in general, but I guess some modification should do.