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 $\mathbb{R}^n$?
contractible open sets
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algebraic-topology
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0Hint: The map $\mathbb R^n\to \mathbb R$ $x\mapsto ||x||$ has the required properties. It is likely that the level curves must be sent to circles around the origin under the homeomorphism (if one exists). – 2012-06-25