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 $\mathbb{R}^n$?
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@Mariano: In my situation the function is not a Morse function and the curves of level are connected by paths, I think that this hypothesis about U are enough, but I can't proof this affirmation. @Anton: I don't know if the Whitehead manifold has a function with this properties. I want to proof that the compact sets $K_m = Uf^{1}([0,m])$ (assuming that $0$ is the minimum value of $f$) are simplyconnected, this will implies that $U$ is simplyconnected to infinite. 


See Hidden convexity (almost, but not quite a duplicate) 

