Homeomorphism of $\mathbb{D}$ and convexity

Asked May 12 '24 at 12:25

Active May 12 '24 at 14:23

Viewed 70 times

I think this should be true.

Let $\varphi: \mathbb{D} \rightarrow \mathbb{D}$ be an homeomorphism such that $\varphi(0) = 0$. Then there exists $r < 1$ such that $\varphi(\mathbb{D}_r)$ is convex. $(\mathbb{D}_r = \{z \in \mathbb{C} \mid |z| \leq r\} )$.

Any ideas?

Edit: If it is false let me lower the conditions a bit. Instead of $\varphi(\mathbb{D}_r)$ being convex, just star-shaped at 0.

Thanks

edited May 12 '24 at 14:23

asked May 12 '24 at 12:25

Alejandro

Sure, take $r = 1$. More seriously, why do you think this is true? What have you tried? – Ben Steffan May 12 '24 at 12:33
I forgot to add r<1. I have the intuition that this should be true and I need it to prove another theorem. – Alejandro May 12 '24 at 12:42
It's like the homeomorphism doesn't move the point 0, and since its continous the points very close to 0 are moved so little that there is a little disk such that the image is convex. – Alejandro May 12 '24 at 12:45
3

The statement is false. I think I can construct an explicit counterexample. – jjagmath May 12 '24 at 13:40
1

Yes, the claim is false. – Moishe Kohan May 12 '24 at 14:09
For diffeomorphisms the answer is positive. – Moishe Kohan May 14 '24 at 19:53

Homeomorphism of $\mathbb{D}$ and convexity

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