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

  • Jean Michel Morel
  • Sergio Solimini
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 14)


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(cD(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.


Universal Constant Projection Property Single Curve Regular Curve Coarea Formula 
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Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Jean Michel Morel
    • 1
  • Sergio Solimini
    • 2
  1. 1.CEREMADEUniversité Paris-DauphineParis Cedex 16France
  2. 2.Srada provinciale Lecce-ArnesanoUniversità degli Studi di LecceLecceItaly

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