# Existence of diffeomorphism

1. Jun 29, 2010

### losiu99

While reading C.C.Pugh's "Real Mathematical Analysis" I've encountered a following statement:

"A starlike set $$U \subset \mathbb{R}^n$$ contains a point $$p$$ such that the line segment from each $$q\in U$$ to $$p$$ lies in $$U$$. It is not hard to construct a diffeomorphism from $$U$$ to $$\mathbb{R}^n$$."

It's little embarassing, but despite my best effort, I cannot figure out how to do it.
I appreciate any help.

2. Jun 29, 2010

### Pere Callahan

U can map the intervall [-pi/2,pi/2] diffeomorphically to R via
$$x\mapsto \tan x$$

You can apply that to every line in your star-shapped set. It's just "blowing up" the set.

3. Jun 30, 2010

### losiu99

I'm still not sure about how to make it work. Assuming bounded set, we can find the "ends" of the line and treat it like ends of [-pi/2, pi/2] interval with $$p$$ as 0, and "blow" the line to infinite length.

I'm more concerned about unbounded sets, where some lines doesn't have "end".

4. Jun 30, 2010

### Landau

Probably U is assumed to be open? If n=2, you can use that star shaped sets (sorry, I am used to "star shaped" instead of "starlike") are simply-connected and then use the Riemann mapping theorem from complex analysis. For general n, I think it is much harder. E.g. see here at MathOverflow, and here. I am curious to hear if Pere Callahan has an easy proof!

5. Jun 30, 2010

### some_dude

I don't get it. If U isn't required to be open, then I don't think it's true. And if it is open, then wouldn't the identity function satisfy this?

6. Jun 30, 2010

### Landau

Yeah, see the first sentence of my post :)
Huh? Are you asserting that the identity map is a diffeo? This is obiously not true since it is isn't even bijective (unless of course U happens to be R^n itself) ...

7. Jun 30, 2010

### some_dude

Ohhh, stupid me, you want the image of the function to be R^n I bet...

Why map P to the origin, and then for ray originating at p approaching a point on the boundary of U, map that to a corresponding ray originating at 0 (= f(p)) approaching infinity. Apply the MVT to projections of the line segments to show it has maximal rank everywhere. Then, I think, you can use the inverse function theorem to show its a diffeo?

8. Jun 30, 2010

### Hurkyl

Staff Emeritus
Then find a way to reduce the general problem to the bounded problem.

It might be worth making sure your method really doesn't work in this case first....

Last edited: Jun 30, 2010
9. Jun 30, 2010

### Pere Callahan

Maybe you can first shrink every direction by applying an arctan function to get something bounded. It should even be possible to contract everything to a ball around the center of the star. Then expand it again. This would certainly (at least I think so) be a bijection. I don't know if there are exotic cases where this function would not be a diffeomorphism.

10. Jul 1, 2010

### losiu99

Thank you very much for your responses. At the first glance it looks like the links contain everything I need. I shall think about concracting-expanding idea as well, since it looks promising.
Thank you all once again.

11. Jul 1, 2010

### Pere Callahan

These links are indeed quite interesting. I'm often surprised about how exotic objects of low dimensional topology can be. Maybe I just know too little about it.