Integral Curvature Rectifiability
Menger curvature can be used to give quantitative conditions for when sets in may be rectifiable. For a Borel measure on a Euclidean space define
- A Borel set is rectifiable if, where denotes one-dimensional Hausdorff measure restricted to the set .
The basic intuition behind the result is that Menger curvature measures how straight a given triple of points are (the smaller is, the closer x,y, and z are to being collinear), and this integral quantity being finite is saying that the set E is flat on most small scales. In particular, if the power in the integral is larger, our set is smoother than just being rectifiable
- Let, be a homeomorphism and . Then if .
- If where, and, then is rectifiable in the sense that there are countably many curves such that . The result is not true for, and for .:
In the opposite direction, there is a result of Peter Jones:
- If, and is rectifiable. Then there is a positive Radon measure supported on satisfying for all and such that (in particular, this measure is the Frostman measure associated to E). Moreover, if for some constant C and all and r>0, then . This last result follows from the Analyst's Traveling Salesman Theorem.
Analogous results hold in general metric spaces:
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