# Uniform Rectifiability and Quasiminimizing Sets of Arbitrary Codimension

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*Guy David; Stephen Semmes*

Roughly speaking, a \(d\)-dimensional subset of
\(\mathbf R^n\) is minimizing if arbitrary deformations of it (in
a suitable class) cannot decrease its \(d\)-dimensional volume. For
quasiminimizing sets, one allows the mass to decrease, but only in a
controlled manner. To make this precise we follow Almgren's notion of
“restricted sets” [2]. Graphs of Lipschitz mappings
\(f\:\mathbf R^d \to \mathbf R^{n-d}\) are always quasiminimizing, and
Almgren showed that quasiminimizing sets are rectifiable. Here we
establish uniform rectifiability properties of quasiminimizing sets, which
provide a more quantitative sense in which these sets behave like Lipschitz
graphs. (Almgren also established stronger smoothness properties under tighter
quasiminimality conditions.)

Quasiminimizing sets can arise as minima of functionals with
highly irregular “coefficients”. For such functionals, one cannot
hope in general to have much more in the way of smoothness or structure than
uniform rectifiability, for reasons of bilipschitz invariance. (See also [9].)

One motivation for considering minimizers of functionals with
irregular coefficients comes from the following type of question. Suppose that
one is given a compact set \(K\) with upper bounds on its
\(d\)-dimensional Hausdorff measure, and lower bounds on its
\(d\)-dimensional topology. What can one say about the structure of
\(K\)? To what extent does it behave like a nice
\(d\)-dimensional surface? A basic strategy for dealing with this issue
is to first replace \(K\) by a set which is minimizing for a measurement
of volume that imposes a large penalty on points which lie outside of
\(K\). This leads to a kind of regularization of \(K\), in which
cusps and very scattered parts of \(K\) are removed, but without adding
more than a small amount from the complement of \(K\). The results for
quasiminimizing sets then lead to uniform rectifiability properties of this
regularization of \(K\).

To actually produce minimizers of general functionals it is
sometimes convenient to work with (finite) discrete models. A nice feature of
uniform rectifiability is that it provides a way to have bounds that cooperate
robustly with discrete approximations, and which survive in the limit as the
discretization becomes finer and finer.

#### Table of Contents

# Table of Contents

## Uniform Rectifiability and Quasiminimizing Sets of Arbitrary Codimension

- Contents vii8 free
- Chapter 0. Introduction 110 free
- Chapter 1. Quasiminimizers 716 free
- Chapter 2. Uniform Rect inability and the Main Result 1019
- Chapter 3. Lipschitz Projections into Skeleta 1322
- Chapter 4. Local Ahlfors-Regularity 2029
- Chapter 5. Lipschitz Mappings with Big Images 2736
- Chapter 6. From Lipschitz Functions to Projections 3039
- Chapter 7. Regular Sets and Cubical Patchworks 3342
- Chapter 8. A Stopping-Time Argument 4453
- Chapter 9. Proof of Main Lemma 8.7 5160
- Chapter 10. Big Projections 7887
- Chapter 11. Restricted and Dyadic Quasiminimizers 8594
- Chapter 12. Applications 97106
- Bibliography 131140