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Let $\alpha$ and $\beta$ $C^1$ boundaries of two topological disks in the plane. We can change $\alpha$ by rigid motions and homoteties to $\tilde{\alpha}$, such that $\beta$ is inside the bounded component of $\tilde{\alpha}$. The point is: "What is the minimal such $\tilde{\alpha}$?" There are lots of questions one can do about this, for example:

"Fix the minimal length of $\tilde{\alpha}$ with that property. How many solutions do we have with this length? We can see this set of solutions as a subset of $\mathbb{R}^2 \times\mathbb{S}^1$."

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  • $\begingroup$ Note that one can pose the problem without talking of the length (it is proportional to the homotety factor). Given the topological disks $A$ and $B$ in the (complex) plane, we want the minimum $r$ such that there holds $zA+w\supset B$ for some complex nunmbers $w$ and $|z| < r$. One could state some optimality conditions for the local minimizers, but these could still be a large set. In the case of starshaped or even convex subsets something more can possibly be said. $\endgroup$ Aug 26, 2012 at 7:37

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