The background to the question is that I would like to figure out how much a map of Europe must distort distances. Let us try to formulate this mathematically.

Say I have a closed subset, $D$, of the unit sphere and I map it bijectively and continuously to a subset $\mathbb{R}^2$ using the function $f$. The function $f$ is said to have distortion $\epsilon$ if $(1-\epsilon) d(f(x),f(x)) \leq d_S(x,y) \leq (1+\epsilon) d(f(x),f(y))$ for all points $x,y \in D$, where $d_S$ is the distance on the sphere and $d$ is the distance in the plane.

How would one find the least possible distortion of $f$ for a given region $D$?

Is there at least a lower bound available on the distortion in terms of some nice property of $D$ like the area or the diameter?


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