Further Properties of Minimizers: Covering the Edge Set with a Single Curve

Abstract

In this final chapter we shall consider further properties of the optimal segmentations. The main new information which we shall obtain is that the whole segmentation K is contained in a single rectifiable curve γ whose length is proportional (with a ratio depending on Ω) to the length of K and which is Ahlfors regular. More precisely, there is a universal constant C such that on every disk D(r), r ≤ 1 one has l(c∩D(r)) ≤ Cr. The existence of such a curve will be obtained as a counterpart to a uniform rectifiability property, stronger than any considered in the last chapter. We shall prove that in every disk centered on K there is a rectifiable curve which is almost equal to a “big piece” of K. This property, which we call “Concentrated Rectifiability Property” will be proved in Sections 1 and 2 by using the “small oscillation covering” technique of Chapters 14 and 15. Section 3 is devoted to the proof that we can somehow join the curves given by this Rectifiability Property and obtain a curve containing all of K. In Section 4, we show by a simple minimality argument that such a curve can be imposed to be Ahlfors regular, with a universal constant only depending on the diameter of Ω.