A few years ago I was doing some research in origami, and was motivated to as the following questions:
Consider $\mathbb{R}^2$ with the Euclidean metric and Lebesgue measure. Does there exist a constant $C > 0$, such that for any connected open set $R \subset \mathbb{R}^2$ with area at least $C$, there exists a surjective contraction $f: R \to [0, 1]^2$?
I managed to give a positive answer to this question when $R$ is convex, but the general case seems elusive to me. I have asked several friends about this, but none of them knew anything like this. I suspect the answer is negative in general, but I have no proof. Specifically, the only "obstruction" I know to this kind of contraction is that the diameter of $R$ must be at least $\sqrt{2}$, but this is automatically satisfies when the area of $R$ is at least $100$.
I want to know: has this kind of problem been researched before? Could anyone point me to a reference?
