The sponge problem

Here’s a cute open problem professor Guth told me about yesterday (also appeared in [Guth 2007], although that version is slightly stronger and more general:

Problem: Does there exist c>0 s.t. \forall U \subseteq \mathbb{R}^2 with m(U) < c, we have \exists f: U \rightarrow \mathbb{R}^2 that’s bi-lipschitz with constant 2 and f(U) \subseteq B_1(0)?

Here bi-lipschitz with constant k is defined in the infinitesimal sense, i.e. \forall p \in U, 1/k < |Df(p)| < k.

i.e. given any open set of a very small measure, say 1/100, does there always exist a map that locally does not deform the metric too much ( i.e. bi-lipschitz with constant 2 ) and “folds” the set into the unit ball.

My initial thought are to approximate the set with some kind of skeleton (perhaps a 1-dimensional set that has U contained in its \delta neighborhood) and fold the skeleton instead. Not sure exactly what to do yet…maybe Whitney’s construction? Or construction similar to creating the nerve in Čech homology? Obviously there is a smaller lipschitz constant allowed when we pass to the skeleton, but that might not be an issue since we can pick the measure of our set arbitrarily small.

By the way, personally I believe the answer is affective…There got to be a way to “fold stuff in” when we don’t have much stuff to start with, right?

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