Denote by $S_r$ the usual circle of radius $r$, with the path metric ($d(x,y) = r\theta$, where $\theta$ is the angle between the vectors $x$ and $y$), and let $\alpha \in (1/2,1)$. Consider the collection of mappings $f: S_r \rightarrow \mathbb{R}^2$ satisfying $|f(x) - f(y)| \leq d(x,y)^\alpha$ for all $x,y \in \mathbb{R}^2$ (no other assumptions on $f$). It makes sense to talk about the area enclosed by $f(S_r)$, basically by taking the integral over $\mathbb{R}^2$ of the winding number relative to $f(S_r)$. Notice that the constraints on $\alpha$ imply that $f(S_r)$ has zero area.

The question then is: what mapping $f$ maximizes this enclosed area? Thinking about the problem for a few minutes would suggest that this would be a circle, as in the classical isoperimetric problem, though with a different radius (which, it seems, is usually impossible to write down explicitly).

The problem seems difficult. The difficulty is that it's inherently global: starting from a given $f$, one can usually expand at a point $x$ without violating the Hölder condition locally, though this might violate the Hölder condition for $x$ relative to a point far away.

I'm curious if it this question has been asked before and whether it would be a tractable problem.


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