Is there a measure of how much a homeomorphism distorts a shape?
In particular, consider the set of orientation preserving homeomorphisms between compact subsets of the euclidean plane. I am looking for a function $d$ from this set to $\mathbb R$ measuring how much $d$ distorts its domain. It should at least satisfy:
- Some consistency properties:
- $d(id) = 0$
- $d(f^{-1}) = d(f)$
- $d(f \circ g) \le d(f) + d(g)$
- $d(f) = 0 \iff f \text { is a similarity}$
Are there any functions like this?