# Least possible distance distortion for a map

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?