Properties of Regular and Rectifiable Sets
The full equivalence between the notions of rectifiability and regularity is developed in this chapter, as well as two very remarkable geometric properties of unrectifiable sets. It is first proved that there is a universal constant c(α) such that at almost every point of a fully unrectifiable set, the lower conic density is larger that c(α). This is a local and clear cut computational criterion for distinguishing rectifiable points from unrectifiable points in a given α-set. From this property, and from the existence almost everywhere of a tangent space for regular sets, it is immediately deduced that fully unrectifiable sets are fully irregular and that regular sets are rectifiable. We also know from Chapter 11 that rectifiable sets are regular. Thus the equivalence between regularity and rectifiability will be complete and, as a by product, the equivalence of simple rectifiability and rectifiability. We end the chapter with the surprising property of (N - 1)-fully irregular (or unrectifiable) sets of ℝN to have a negligible projection on almost every hyperplane.
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