Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\mathbb{R}^n$?

share|improve this question
    
Hint: 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). –  Simon Markett Jun 25 '12 at 17:23

1 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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.