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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$\begingroup$ If the function were a Morse function (in particular, smooth) this would follow from Morse theory—maybe that is enough for your? (Could the level sets not be path connected in your situation?) $\endgroup$– Mariano Suárez-ÁlvarezJun 25, 2012 at 17:57
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$\begingroup$ @Mariano: You can make a spiral construction in $\mathbb{R}^2$ such that that the inverse of a value (say 1) is the unit circle and some curves in the unit disc spiraling out towards the unit circle, but never getting there. You can put the unique minimum at 0 taking the value 0. The inverse image of 1 would be connected but not path connected. $\endgroup$– Thomas KraghJun 25, 2012 at 18:21
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2$\begingroup$ If $U$ contains the origin, then $f=x_1^2+...+x_n^2$ satisfies the first condition and is a Morse function, but the set is not necessarily homeomorphic to $\mathbb R^n$. ( math.stackexchange.com/questions/55114/… ) Level sets can easily fail to be path-connected, for instance take $x_1^2+...x_{n-1}^2<1$, then '$\{f>1\}$' has two components. $\endgroup$– Will SawinJun 25, 2012 at 18:22
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$\begingroup$ You should assume your function to be proper. Perhaps, in case of not differentiability, please assume also that, away from the minimum point, $f$ is locally homeomorphic to the map $(x_1, \dots, x_n) \mapsto x_1$. This should suffices. $\endgroup$– Daniele ZuddasJun 25, 2012 at 18:27
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8$\begingroup$ What about Whitehead manifold? $\endgroup$– Anton PetruninJun 25, 2012 at 18:36
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2 Answers
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See Hidden convexity (almost, but not quite a duplicate)
answered Jun 26, 2012 at 0:53
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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 $K_m = U-f^{-1}([0,m])$ (assuming that $0$ is the minimum value of $f$) are simply-connected, this will implies that $U$ is simply-connected to infinite.
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$\begingroup$ This should really have been a comment below Mariano's answer (well, the comment to Anton would have gone in the comment thread below the answer). There is an extensive wikipedia page on the Whitehead manifold. It is an open 3-manifold which is contractible but not homeomorphic to $\mathbb{R}^3$. You'd still have to find a map from it to $\mathbb{R}$ with only one minimum, but basically Anton was giving evidence that the answer to your question is no. The terminology "simply-connected to infinity" is not standard, but does appear in the SE thread Will Sawin linked to $\endgroup$ Jul 24, 2012 at 12:44