BOUNDARY MEASURES and CUBICAL COVERS of SETS in Rn

BOUNDARY MEASURES AND CUBICAL COVERS

OF SETS IN Rn

LARAMIE SMITH PAXTON

A dissertation submitted in partial fulﬁllment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

WASHINGTON STATE UNIVERSITY Department of Mathematics and Statistics

JULY 2018

The members of the Committee appointed to examine the dissertation of LARAMIE SMITH PAXTON ﬁnd it satisfactory and recommend that it be accepted.

Kevin R. Vixie, Ph.D., Chair

Bala Krishnamoorthy, Ph.D.

Matthew Sottile, Ph.D.

ii ACKNOWLEDGEMENT

I would like to acknowledge Kevin R. Vixie for his boundless support regarding this project. Without his patience and encouragement, this work would not have been possible.

iii BOUNDARY MEASURES AND CUBICAL COVERS

OF SETS IN Rn

Abstract

by Laramie Smith Paxton, Ph.D. Washington State University July 2018

Chair: Kevin R. Vixie

The art of analysis involves the subtle combination of approximation, inequalities, and geometric intuition as well as being able to work at different scales. Even the restriction to sets in Rn affords us the opportunity to hone these skills through a virtually limitless supply of examples and tools all aimed at “taming the wild,” so to speak, so as to develop techniques by which to work with arbitrary sets. In this dissertation, we present a singular integral for measuring the level sets (i.e. the boundary of the super-level or sub-level set) of a C1,1 function mapping from Rn to R, that is, one whose derivative is Lipschitz continuous. We extend this to measure embedded manifolds in R2 that are merely C1 using the distance function. We then present a rather playful exposition, with several open problems, related to representations of sets in Rn aimed at stimulating interest and inspiring student research in these areas. We explore finding less and less regular sets for which it is still true that the representations faithfully inform us about the original set without removing important features. While the primary focus is on cubical covers, we also briefly introduce Jones’ β numbers and varifolds from geometric measure theory, and we present a conjecture for the

iv ratio of the measure of the boundary of the cubical cover to the measure of the C1,1 boundary of a compact set.

v TABLE OF CONTENTS

Page

ACKNOWLEDGEMENT ...... iii

ABSTRACT ...... iv

LIST OF FIGURES ...... viii

1 INTRODUCTION ...... 1

1.1 Overview...... 1

1.2 A Simple Analysis Problem...... 3

1.3 Higher Dimensions...... 8

1.4 Lipschitz Continuous...... 10

1.5 Maximal Growth Rates & Metric Spaces...... 12

2 BACKGROUND INFORMATION ...... 15

2.1 A Brief Description of Singular Integrals...... 15

2.2 Geometric Measure Theory...... 16

3 A C1,1-BOUNDARY MEASURE ...... 22

3.1 Measuring Level Sets...... 22

3.2 A Simple Example...... 29

4 A C1-BOUNDARY MEASURE ...... 33

4.1 The n = 2 Case...... 33

5 CUBICAL COVERS OF SETS IN Rn ...... 45

5.1 Representing Sets & their Boundaries in Rn ...... 45

5.2 Cubical Reﬁnements: Dyadic Cubes...... 45

vi 5.3 Jones’ β Numbers...... 47

5.4 Working Upstairs: Varifolds...... 48

5.5 Simple Questions...... 49

5.6 A Union of Balls...... 50

5.7 Minkowski Content...... 53

5.8 Smooth Boundary, Positive Reach...... 56

5.9 A Boundary Conjecture...... 60

5.10 The Jones’ β Approach...... 62

5.11 A Varifold Approach...... 65

5.12 Problems and Questions...... 67

6 SUMMARY ...... 71

APPENDIX ...... 74

A.1 Further Exploration...... 75

A.2 Measures: A Brief Reminder...... 76

BIBLIOGRAPHY ...... 80

vii LIST OF FIGURES

1.1 Constructing fences I...... 4 1.2 Constructing fences II...... 5 1.3 Constructing fences III...... 5

1 1.4 The cylinder CS...... 11

2.1 Hausdorﬀ measure...... 17 2.2 The Area Formula...... 19

3.1 A diﬀeomorphism H : C → E...... 24 3.2 Bounding the derivative at y...... 26 3.3 The graph of f = x2 + y2 − 1...... 29

4.1 Global distance of A to itself...... 33

4.2 Covering A with disks B(xi, δ)...... 34 4.3 Taking the limit inside B(0, 2R)...... 35

η 4.4 Covering A with rectangles Ei ...... 36

η 4.5 A ∩ Ei is contained in Ki ...... 37 η 4.6 Overlaying A with Ei ’s...... 38

∗ 4.7 Translating Ci along A ...... 41 4.8 Bounds of integration...... 42

5.1 Dyadic cubes...... 46

E 5.2 Cubical cover Cd of a set E...... 47 5.3 Jones’ β numbers...... 48 5.4 The vertical axis for the “upstairs.”...... 49 5.5 Working upstairs in the Grassmann bundle...... 49 5.6 Concentric cubes...... 51

viii 5.7 Minkowski content...... 54 5.8 Positive and non-positive reach...... 57 5.9 Moving out and sweeping in...... 57 5.10 Cubes on the boundary...... 60

5.11 The angle between Tx∂E and the x-axis...... 62

5.12 Jones’ β numbers and WC ...... 63 5.13 Working upstairs...... 66 5.14 Multiscale ﬂat norm decomposition...... 69

A.1 Approximate tangent planes I...... 78 A.2 Approximate tangent planes II...... 79

ix CHAPTER 1

INTRODUCTION

1.1 Overview

Often times it happens that creative exploration leads to uncharted territory in mathematics. That is indeed the case in the development of the singular integral boundary measure for hypersurfaces in Rn, presented below, and the exposition of cubical covers of sets in Rn that follows. To highlight this process, we shall first demonstrate the investiga- tions leading up to our discovery of this integral and its properties from the starting point of a simple analysis problem in R. We then extend the result of this problem to differentiable functions mapping from Rn to Rm and then to Lipschitz continuous functions (Definition 1.4.1), which are differentiable almost everywhere by Rademacher’s Theorem [20]. After further examination of the ratio ||Df||/||f||, we present a singular integral that measures the (n − 1)-dimensional Hausdorff measure Hn−1 of level sets of C1,1 functions, that is those whose derivative is Lipschitz continuous. Continuing the process of exploration and generalization, we present the same singular integral as a C1-boundary measure using the distance function in R2. The arguments here are markedly different from the C1,1 case and follow a “by-construction,” barehanded approach due to the fact that in this case, the derivative of the distance function is not Lipschitz (so we cannot use the Area Formula; see Theorem 2.2.2) and we may have -neighborhoods of the boundary in which the normals intersect no matter how small is (i.e. we no longer have positive reach; see Definition 5.8). Lastly, we present an exposition of cubical covers geared toward engaging undergradu- ates in analysis research. For example, while the properties of Radon measures offer a more direct approach to answering the questions we pose, our approach provides a more instructive and informative introduction to the ideas. Using covers of n-cubes, we investigate bounds

1 for various types of sets in Rn as well as their boundaries. Let Ln be the n-dimensional

E Lebesgue measure, and deﬁne Cd as the union of n-cubes parallel to the coordinate axes

d n E with edge length 1/2 that intersect a set E ⊂ R , letting the boundary ∂Cd be the topo- logical boundary of this union. The overall theme is under what conditions can we say the following?

n E n (1.) There exists a d0 such that for all d ≥ d0, we have L (Cd ) ≤ M(n)L (E), for some constant M(n) independent of E.

n E n (2.) For all δ > 0, there exists a d0 such that for all d ≥ d0, we have L (Cd ) ≤ (1 + δ)L (E).

Using cubical covers, we then show the following:

• (1.) holds if E ⊂ Rn is locally Ahlfors n-regular, meaning there exists an M ≥ 1

−1 n and an r0 > 0 such that for all x ∈ E and for all 0 < r < r0, we have M r ≤ Ln(B(x, r) ∩ E) ≤ Mrn, where B(x, r) is the ball of radius r centered at x.

• (2.) holds if E ⊂ Rn is such that Hn−1(∂E) < ∞ and ∂E is an (n − 1)-rectiﬁable set, meaning there is a Lipschitz function mapping a bounded subset of Rn−1 onto ∂E (see Deﬁnitions 1.4.1, 2.2.1, and 5.7.3).

• (2.) holds if E ⊂ Rn is such that ∂E is smooth enough (at least C1,1) to have positive reach (see Section 5.8).

E We next present a conjecture for the first question above regarding ∂E and ∂Cd with a sketch of the proof for n = 2 along with several open problems, one of which involves the decomposition of sets using a special norm called the multiscale flat norm prior to finding the cubical cover approximation (see Section 5.12). In addition, we briefly introduce another representation of sets in Rn, Jones’ β numbers, as a means of determining if a set E ∈ Rn lies in a rectifiable curve Γ (Theorem 5.10.1). Intuitively, we can think of this as determining how “flat” E looks as we zoom in on it since Γ, being rectifiable, has a tangent at H1-almost every point x ∈ Γ.

2 Lastly, we present an introduction to varifolds from geometric measure theory (Deﬁni-

tion 5.11.2) which, by attaching the m-dimensional Grassmannian to each point in Rn (i.e. the set of m-dimensional tangent planes through the origin in Rn), encapsulates information about the tangent space. We also include in the appendix a detailed section on additional resources for those interested in geometric measure theory and some of the current research being done in this area.

1.2 A Simple Analysis Problem

Consider the following simple analysis problem:

Problem 1.2.1. For f(x): R → R, suppose that f(x) is diﬀerentiable, f(0) = 0, and ∃λ > 0 such that ∀x ∈ R

|f 0(x)| ≤ λ|f(x)|. (1.1)

Prove that f(x) = 0 everywhere.

2 2λx −2λx Solution 1: ODE’s. Take the families of curves in R gC = Ce and hC = Ce as all the solutions to the diﬀerential equations g0 = 2λg and h0 = −2λh, respectively. Cleary f(x) cannot simply be one of these curves since none of them are ever 0. Note that the 2 is included here so that any solution f(x) that satisﬁes (1.1) that intersects one of these curves

must cross it. Indeed, without loss of generality, suppose for gC there exists an x0 such that

2λx for all x > x0, f(x) = C0e . Then we ﬁnd that

2λx 2λC0e 2λx ≤ λ ⇒ 2 ≤ 1, C0e

a contradiction. A similar argument holds for hC .

Now, for gC , f(x) can only cross from above to below as x increases. If not, then

0 2λx f 0 f (x) > C2λe = 2λf ⇒ f > 2λ, also a contradiction. Similarly, f(x) can only cross hC

3 from below to above as x increases. Now, suppose f(x∗) > 0 for some x∗ (for f(x∗) < 0, then −f(x) satisﬁes (1.1) and is

∗ +∗ ∗ 2λx∗ −∗ ∗ −2λx∗ positive at x ). Pick the corresponding C = f(x )/e for gC+∗ and C = f(x )/e for hC−∗ . These curves serve as “fences” (gC+∗ for x < x∗ and hC−∗ for x > x∗) that f cannot cross so that f 6= 0. Thus we have a contradiction which implies that f(x) ≡ 0 everywhere (see Figures 1.1, 1.2, and 1.3).

gC+∗

f(x∗)

x∗

∗ Figure 1.1: The case where f(x ) > 0 and gC+∗ acts as a “fence” for f(x) when x < x∗, bounding it away from 0.

4 hC−∗

f(x∗)

x∗

∗ Figure 1.2: The case where f(x ) > 0 and hC−∗ acts as a “fence” for f(x) when x > x∗, bounding it away from 0.

hC−∗ gC+∗

f(x∗)

x∗

Figure 1.3: The two combined cases from above.

5 Solution 2: Mean Value Theorem. Suppose f(x0) = 0. We prove f(x) = 0 on I = [x0 −

1 1 2λ , x0 + 2λ ]. Cleary then we can pick the endpoints of this interval to extend our interval where f(x) is 0. Continuing this process thus this implies that f(x) is 0 everywhere. Let x ∈ I. By the Mean Value Theorem,

1 |f(x) − f(x )| ≤ |f 0(y )||x − x | ≤ |f 0(y )| , 0 1 0 1 2λ

for some y1 ∈ I. By (1.1), we have

1 1 |f(x)| = |f(x) − 0| ≤ |f 0(y )| ≤ f(y ). 1 2λ 2 1

1 1 Similarly, |f(y1)| ≤ 2 f(y2) for some y2 ∈ I. Thus |f(x)| ≤ 22 f(y2). Continuing this process, we obtain 1 |f(x)| ≤ f(y ) (1.2) 2n n for some yn ∈ I, n ∈ N. From the Extreme Value Theorem, f(x) continuous ⇒ f(x) < M for all x ∈ I. Using this with (1.2), we get M |f(x)| ≤ for all n ∈ . 2n N

Sending n → ∞ gives f(x) ≡ 0.

Solution 3: A Barehanded ODE Approach. We shall construct “fences” similar to the ﬁrst solution above. If f(x) > 0 ∀x ∈ [x0, x1] ⊂ E ⊂ R, then

f 0(x) |f 0(x)| ≤ λ|f(x)| ⇔ −λ ≤ ≤ λ. f(x)

6 Integrating both sides of this over the interval [x0, x1] gives

f(x1) −λ(x1 − x0) ≤ ln ≤ λ(x1 − x0). f(x0)

But this is equivalent to

f(x1) e−λ(x1−x0) ≤ ≤ eλ(x1−x0). (1.3) f(x0)

Now, let f(x∗) > 0. Deﬁne

l = inf{w|f(x) > 0 for all w < x < x∗} and u = sup{w|f(x) > 0 for all x∗ < x < w}.

We see that l 6= −∞ ⇒ f(l) = 0 and u 6= ∞ ⇒ f(u) = 0.

∗ Using (1.3), assume l 6= −∞. Without loss of generality, let x1 = x and pick a sequence

{x0} ↓ l. In the limit we ﬁnd that ∞ ≤ C, where C is ﬁnite; hence a contradiction.

∗ Or, we may assume u 6= ∞. Let x0 = x and pick a sequence {x1} ↑ u to obtain a similar contradiction of C ≤ 0, where C is positive. In either case, we find that there is no x∗ such that f(x∗) > 0. Recalling that if f(x∗) < 0, then −f(x) satisfies (1.1) and is positive at x∗, we find that f(x) = 0 everywhere.

One interpretation of what we have shown so far is if

1. f is diﬀerentiable,

2. f(x0) = 0, and

3. for some δ > 0, we have that f(x) 6= 0 when x 6= x0 and x ∈ [x0 − δ, x0 + δ],

0 then letting A (x) ≡ f (x) , f f(x)

7 lim sup Af (x) = ∞. x→x0

0 Hence we see quite clearly that the ratio f (x) detects roots. We shall rely on this property f(x) in our singular integral boundary measure introduced in the next chapter. Further, if we deﬁne

a(f) ≡ sup Af (x), x∈R we ﬁnd that

a(f) = a(αf) for all α 6= 0.

0 Thus scaling f by non-zero constants does not aﬀect sup f (x) . x f(x)

1.3 Higher Dimensions

It is the method of Solution 3 above, especially inequality (1.3), that we shall now use to generalize this simple analysis problem from being diﬀerentiable in one dimension to see that the same result holds in higher dimensions and if f is merely Lipschitz continuous (see Deﬁnition 1.4.1).

Deﬁnition 1.3.1. Given two normed vector spaces V and W (over R or C), let T : V → W be a bounded linear operator. We deﬁne the operator norm as

||T ||op = inf{c ≥ 0 : ||T v|| ≤ c||v|| for all v ∈ V },

where || · || is the Euclidean norm. We see that the operator norm tells us the maximum amount that T “stretches” vectors.

Now, let B(x, r) be the open ball of radius r centered at x ∈ Rn, and let f : Rn → Rm.

8 Then (1.1) becomes ||Df(x)|| ≤ λ||f(x)||, where ||Df(x)|| is the operator norm of the diﬀerential, or derivative, of f(x), Df(x), and ||f(x)|| is the Euclidean norm of f(x). Notice

1 f(x) T D ln(||f(x)||) = Df(x), ||f(x)|| ||f(x)|| where T is the transpose.

Let γ(s) be the arclength parameterized line segment that starts at x0 and ends at x1. Then we have

Z Z f(x(s)) T Df(x(s))ds Z ||Df(x(s))||ds D ln(||f(x(s))||)ds = ≤ . γ γ ||f(x(s))|| ||f(x(s))|| γ ||f(x(s))||

Thus ln(||f(x1)||) − ln(||f(x0)||) ≤ λ||x1 − x0||, which implies that

||f(x1)|| −λ||x1 − x0|| ≤ ln ≤ λ||x1 − x0||, ||f(x0)||

and we can proceed as in Solution 3 above, using surfaces as “fences” instead of curves. Loosely speaking, the graph of f lives inside the “cones” deﬁned by these exponentials. More precisely, we ﬁnd that if ||Df(x)|| ≤ λ||f(x)|| and ||f(x)||= 6 0 for all x ∈ B(x∗, r), then

∗ ||f(x)|| ∗ e−λ||x−x || ≤ ≤ eλ||x−x || ||f(x∗)||

for all x ∈ B(x∗, r), which implies that if f(x) = 0 anywhere, it equals 0 everywhere.

Remark 1.3.1. Let ||Df(x)|| be the operator norm of the Fr´echetderivative Df(x) and let

||f(x)|| the B2-norm of f(x) ∈ B2. Then the result just shown holds for f : B1 → B2, where

B1 and B2 are Banach Spaces.

9 1.4 Lipschitz Continuous

Deﬁnition 1.4.1. A function f : Rn → Rm is Lipschitz continuous if ∃α > 0 such that

n |f(x) − f(y)| ≤ α|x − y| for all x, y ∈ R . We will call L ≡ infβ{αβ} the Lipschitz constant of f.

Note that Rademacher’s Theorem states that a (locally) Lipschitz function f : Rn → Rm is diﬀerentiable Ln-almost everywhere [20], where Ln is the n-dimensional Lebesgue measure.

Now, let f : Rn → Rm be Lipschitz. In this case, the above argument must be adapted since f may not be diﬀerentiable anywhere on the line segment from x0 to x1, which we shall denote by S.

1 Let CS be a cylinder of radius 1 with its axis S, and let

1 E ≡ CS ∩ {x|Df(x) exists }.

Then we see that

n 1 f diﬀerentiable a.e. ⇒ L (CS \ E) = 0,

1 which implies that almost every line segment in CS parallel to S and of length ||x1 − x0|| intersects E in a set of length ||x1 − x0||. Thus we can choose a sequence of such segments

{Sk} → S.

10 1 X1 2 X1 n X1

1 X0

2 X 0 n X0 X0

1 Figure 1.4: The cylinder CS with full length line segments converging to its axis S, the segment from x0 to x1. (Image used with permission from Kevin R. Vixie.)

1 Since Df exists L -a.e. on the segments Sk and f is continuous everywhere, we can integrate the derivatives as above to get

k k k ||f(x1)|| k k −λ||x1 − x0|| ≤ ln k ≤ λ||x1 − x0||. ||f(x0)||

And because f is continuous, we get

||f(x1)|| −λ||x1 − x0|| ≤ ln ≤ λ||x1 − x0||, ||f(x0)|| which immediately implies the same result that we had for diﬀerentiable functions above.

11 1.5 Maximal Growth Rates & Metric Spaces

Recall our deﬁnition above for

||Df(x)|| a(f) ≡ sup , x∈R ||f(x)||

and the fact that a(f) = a(αf) for all α 6= 0. (1.4)

Let C1(R, R) denote the continuously differentiable functions from R to R. If we define C = {f | a(f) ≤ λ}, we find that not only is S C not all of C1( , ), we also have λ n∈Z+ n R R functions satisfying 0 < b ≤ f(x) ≤ B < ∞ whose a(f) = ∞. So, we shall restrict this class of functions to the space of continuously differentiable functions from K ⊂ R to R, C1(K, ), where K = [−m, m]. Now, C \ S C contains only those functions which R ∞ n∈Z+ n have a root in K.

1 We will call the functions in Cλ ⊂ C (K, R) functions with maximal growth rate λ. This is a natural moduli for functions that describe systems whose maximal growth rate depends linearly on the current value, of which populations are the best example. Based on the results above, we know that if f ∈ Cλ, then its graph is enclosed in a “cone” deﬁned by exponentials. That is, if a(f) = λ, then for x < x0,

f(x ) f(x ) 0 eλx ≤ f(x) ≤ 0 e−λx, eλx0 e−λx0

and for x > x0 we have f(x ) f(x ) 0 e−λx ≤ f(x) ≤ 0 eλx. e−λx0 eλx0

Remark 1.5.1. Notice that (1.4) and λ < ∞ imply that scaling a function in Cλ by any

nonzero scalar yields another function in Cλ. As a result, we could consider only

12 F ≡ {f ∈ Cλ such that f(0) = 1},

F ≡ { functions whose minimum value on K is 1}.

In both cases we end up with subsets that generate Cλ when we take all scalar multiples of those functions.

n ||Df|| Remark 1.5.2. Let E ⊂ R . We note here a peculiarity of the ratio ||f|| : If f(x) = Ln(E(x)), where x can be thought of as the center of the set, we have that Df will be a vector field η times Hn−1 (see Definition 2.2.1) restricted to ∂E(x). Thus, ||Df|| will be an (n−1)-dimensional quantity and f a n-dimensional quantity. As in the case of the Poincaré inequality, we could add exponents to make the ratio non-dimensional.

As another example of generalizing our simple analysis problem in (1.1), suppose that

X is a metric space and f : X → R. Further suppose that γ : R → X is continuous and is a geodesic in the sense that for any three points in R, s1 < s2 < s3, we have that

ρ(γ(s1), γ(s3)) = ρ(γ(s1), γ(s2)) + ρ(γ(s2), γ(s3)). If we have that

1. for any two points in the metric space there is a γ containing both points;

2. for all such γ, gγ ≡ f ◦ γ is diﬀerentiable;

0 |gγ (s)| 3. and |f(γ(s))| ≤ λ, then we ﬁnd

|f(x1)| − λρ(x1, x0) ≤ ln ≤ λρ(x1, x0), (1.5) |f(x0)| as before.

13 Remark 1.5.3. We may also start with any metric space and consider curves γ :[a, b] ⊂

R → X for which

n−1 X l(γ) ≡ sup ρ(γ(si), γ(si+1)) ≤ ∞, n {{si}i=1|a=s1≤s2≤...≤sn=b} i=1 that is, rectiﬁable curves. We can always reparameterize such curves by arclength so that γ(s) = γ(s(t)), t ∈ [0, l(γ)] and l([γ(s(d)), γ(s(c))]) = d − c. We will assume that all curves have been reparam- eterized by arclength. Now, we deﬁne a new metric that will not change the length of any curve.

ρ˜(x, y) = inf l(γ). {γ|γ(a)=x and γ(b)=y}

Deﬁne an upper gradient of f : X → R to be any non-negative function ηf : X → R R such that |f(y) − f(x)| ≤ γ ηf (γ(t))dt.

|ηf (x)| Now, if |f(x)| ≤ λ, we again get a similar set of bounds as in (1.5) if we replace ρ with ρ˜. See [25] for more about upper gradients.

Thus far, the above explorations, while yielding no significantly new results, have laid out a path of discovery that students of mathematics (young and old) may emulate to facilitate a more robust, more useful grasp of a rather simple initial solution. This is a highlight of the process that we undertook to not only “generalize” Problem 1.2.1, but to probe the exact meaning of the result. In doing so, we discovered deeper properties of the ratio ||Df||/||f|| that led to the singular integral in Theorem 3.1.1. Before presenting this theorem, we shall briefly introduce singular integrals, in general, and present several necessary background definitions and theorems from geometric measure theory and analysis.

14 CHAPTER 2

BACKGROUND INFORMATION

2.1 A Brief Description of Singular Integrals

Singular integrals are central to harmonic analysis [53] and are properly classiﬁed as a type of integral transform:

Z T (f)(x) ≡ K(x − y)f(y) dy,

which is more generally a type of operator. In this case, we input a function f and output

T (f), which is speciﬁed by a kernel function K : Rn × Rn → R that has a singularity anywhere x = y, where x, y ∈ Rn. True singular integrals in this sense are such that |K(x, y)| is of size |x−y|−n as |x−y| → 0 so that we may deﬁne them rigorously using the Cauchy Principal Value (p.v.) method

over the set in Rn where |x − y| > and then let go to 0. This process allows convergence of an otherwise divergent integral due to the cancellation of positive and negative values in the integral. We note here that our singular integral presented below does not converge according to this method although we do employ appropriate limiting processes to obtain convergence, so the similarities are worth noting. The most well known examples of singular integrals in classical analysis are the Hilbert Transform, 1 Z f(y) H(f)(x) = lim dy, π ε→0 |x−y|≥ε x − y

and its higher dimensional analogues, the Riesz Transforms. The singular integrals mentioned thus far are of convolution type, and in the case of the Hilbert Transform, we see that it is a convolution of f against the kernel function 1/(πx).

Recall that the convolution of two integrable functions f and g mapping from Rn to R

15 is defined as Z (f ? g)(t) = f(s)g(t − s)ds, Rn which we can think of as the weighted average of f(s) at the moment t with the weight g(t − s). Since g emphasizes different parts of f over different times t, convolution acts as a “smoothing operator” that brings out the “best” properties of both functions f and g.

2.2 Geometric Measure Theory

In this section, we recall some important deﬁnitions and results for later use. In the next chapter, we present the ﬁrst main result regarding the Hn−1-measure of the 0-level sets of C1,1 functions and compute the singular integral (3.1) for the simple example of a paraboloid whose 0-level set is the unit circle.

Deﬁnition 2.2.1. Hausdorﬀ Measure. With this outer (radon) measure, we can measure

k-dimensional subsets of Rn (k ≤ n). While it is true that Ln = Hn for n ∈ N (see Section 2.2 of [20]), Hausdorﬀ measure Hk is also deﬁned for k ∈ [0, ∞) so that even sets as wild as fractals are measurable in a meaningful way (see Figure 2.2).

To compute the k-dimensional Hausdorﬀ measure of A ⊂ Rn:

∞ 1. Cover A with a collection of sets E = {Ei}i=1, where diam(Ei) ≤ d ∀i.

2. Compute the k-dimensional measure of that cover:

k X diam(Ei) Vk(A) = α(k) , E 2 i

where α(k) is the k-volume of the unit k-ball.

k k 3. Deﬁne Hd(A) = infE VE (A), where the inﬁmum is taken over all covers whose elements have maximal diameter d.

k k 4. Finally, we deﬁne H (A) = limd↓0 Hd(A).

16 k P diam(Ei) i α(k) 2

diam(E)

Figure 2.1: The Hausdorﬀ measure is derived from a cover of arbitrary sets. (Image used with permission from Kevin R. Vixie.)

Recall that from Deﬁnition 1.4.1, a function f : Rn → Rm is Lipschitz continuous if

n ∃α > 0 such that |f(x) − f(y)| ≤ α|x − y| for all x, y ∈ R , and we set L ≡ infβ{αβ} to be the Lipschitz constant of f. Also, recall that Lipschitz functions are diﬀerentiable almost everywhere by Rademacher’s Theorem. Thus, while a C1 function is one which is continuously diﬀerentiable, a C1,1 function is one whose derivative is Lipschitz continuous. By a “C1,1 set E,” we shall mean that for

n all x ∈ ∂E, there is a neighborhood of x, Ux ⊂ R , such that after a suitable change of

1,1 n−1 coordinates, there is a C function f : R → R such that ∂E ∩ Ux is the graph of f.

Definition 2.2.2. [See also Definition 5.8.1.] The reach of E, reach(E), is defined

reach(E) ≡ sup{r | every ball of radius r touching E touches at a single point}.

Remark 2.2.1. If a set E has positive reach, then there exist -neighborhoods U of E for all < reach(E) such that for every y ∈ U not in E, there exists a unique closest point x ∈ E, which is to say the normals ~n(x) to E don’t intersect in any U, where < reach(E).

Remark 2.2.2. If ∂E is C1,1, then E has positive reach (see Remark 4.20 in [22]).

17 Theorem 2.2.1. [32][35] (Implicit Function Theorem) Let f : Rn × Rm → Rm be a

k,1 C function, k ≥ 1 with the property that f(x0, y0) = 0 and suppose that ∇yf(x0, y0) 6= 0.

Then there exists an open neighborhood U of x0 and an open neigborhood V of y0 such that for any x ∈ U there exists a unique g(x) ∈ V with the property that f(x, g(x)) = 0. The

k,1 function g : U → V satisﬁes g(x0) = y0 and g ∈ C .

The next theorem is a generalization of the change of variables formula from calculus, the ﬁrst part of which says if f behaves nicely enough, that is, f is Lipschitz, then we can calculate the Hn-measure of f(A) under suitable conditions.

Theorem 2.2.2. [20] (Area Formula I) Let f : Rn → Rm be Lipschitz continuous, n ≤ m, and dx ≡ dLnx. Then for each Ln-measurable subset A ⊂ Rn,

Z Z Jfdx = H0(A ∩ f −1{y})dHny. A Rm

Moreover, if Ω is an Ln-measurable subset of Rn, then for each Ln Ω-integrable function g(x), Z Z X g(x)Jf dx = g(x)dHny. Ω f(Ω) x∈f −1{y}

Here Jf is the Jacobian of f, the n-volume expansion/contraction factor associated with the linear approximation Df at each point in the domain of f, and since f is Lipschitz, it is diﬀerentiable Hn-a.e. so that Jf exists Hn-a.e. The multiplicity function H0(A ∩ f −1{y}) takes into account the case in which f is not 1-1 (see Figure 2.2).

18 2 3 R R

A f(A)

Figure 2.2: Mapping f such that the magenta areas correspond to multiplicity 2 regions in the image f(A). Note that both cases show the domain of the set and not the graph space.

Since Lipschitz maps cannot increase the Hausdorff dimension of A, we can disregard the case where f is a space-filling curve. Clearly though, the Hn-measure of f(A) would be infinite, as a plane has infinite length, for instance. On the other hand, if Df is singular, then Jf is identically 0, so for example, in the case of a 2-d set being compressed into a 1-d set, we have information being lost, and the Hn-measure of f(A) is 0, as expected.

Applications of the Area Formula include the case of mapping a curve from R into R3 and ﬁnding its new length; computing the surface area of a graph or of a parametric hypersurface; or ﬁnding the volume of a set mapped into a submanifold. As a simple example, we present

the case of ﬁnding the length of a portion of the graph of the sine function in R.

Example 2.2.1. Since f = sin(x) is Lipschitz, we let the collection of points g(x) = (x, sin(x)), x ∈ [0, 2π] deﬁne the graph of f over [0, 2π]. We compute the arc length (i.e., the 1-d surface area) of f on the set [0, 2π].

  1   Dg =   cos(x)

Since g is 1-1 and Jg = p1 + cos2(x), we have H1(Graph of f on [0, 2π])= (arc length

R 2π p 2 of f on [0, 2π]) = 0 1 + cos (x) dx ≈ 7.64.

19 We next present a special case of the second part of the Area Formula stated above.

Theorem 2.2.3. [47] (Area Formula II) Let f : Rn → Rm be one-to-one and Lipschitz continuous, n ≤ m, and dx ≡ dLnx. If Ω is an Ln-measurable subset of Rn, then for each Ln Ω-integrable function g(x),

Z Z g(x)Jf dx = g(f −1(y))dHny. (2.1) Ω f(Ω)

Definition 2.2.3. Let f : Rn → Rm be a differentiable function with m ≤ n.A regular value of f is a value c ∈ Rm such that the differential Df is surjective at every preimage of c. This implies that Df is full rank on the set f −1{c}.

We are interested in here in the case m = 1, which if c is a regular value implies ||Df||= 6 0 on f −1{c}. In fact, if we let f ∈ C1 and f −1{c} ⊂ B(0,R) (closed), we claim that there exists a neighborhood E of f −1{c} on which ||Df|| > 0 for all y ∈ E. Indeed, since

0 −1 Df ∈ C we have 0 < < ||Dxf|| < M < ∞ for all x ∈ f {c}.

Let W ≡ {y | ||D f|| < }. Now, suppose there exists a distinct sequence {y } ⊂ W 2 y 2 i 2 −1 1 such that d(yi, f {c}) < 2i , i = 1, 2,.... Since yi ∈ B(0, 2R), by the Bolzano-Weierstrass ∗ ∗ Theorem, there exists a subsequence {yik } and a point y such that yik → y as k → ∞.

∗ −1 −1 Since d(y , f {c}) = 0, we have {yik } → f {c}.

∗ But ||Df|| is continuous, so this implies that ||Dy f|| < 2 , which is a contradiction. −1 Thus there exists a δ > 0 such that d(f {c},W ) > δ, which implies 2

∪ −1 (B(x, δ) ∩ W ) = ∅. x∈f {c} 2

The result follows by taking ∪x∈f −1{c}B(x, δ) to be the neighborhood where ||Df||= 6 0.

Remark 2.2.3. The general case of showing Df is full rank in a neighborhood of f −1{c} for f as in Deﬁnition 2.2.3 can be shown using the continuity of the determinant of a non-zero

square submatrix in Dcf of maximal dimension, which exists because Dcf is full rank.

20 Theorem 2.2.4. [28] Regular Value Theorem: If c ∈ M is a regular value of a diﬀeren- tiable map f : N → M (where N and M are n- and m- dimensional manifolds, respectively), then its preimage f −1{c} ⊂ N is a submanifold whose codimension is equal to the dimension of M.

We note that f −1{c} is also a closed set since f ∈ C(Rn) implies that the inverse image of a closed set is closed.

21 CHAPTER 3

A C1,1-BOUNDARY MEASURE

3.1 Measuring Level Sets

Theorem 3.1.1. Let f : Rn → R1, and let B(0,R) represent the closed ball of radius R centered at the origin. If 0 ∈ f(Rn) is a regular value of f ∈ C1,1 and f −1{0} ⊂ B(0,R), then ( k−1 ) 1 Z ||Df|| k lim dx = 2Hn−1({x | f(x) = 0}). (3.1) k→∞ k B(0,2R) |f|

Proof. First, note that || · || is the operator norm (see Deﬁnition 1.3.1), which in this case corresponds precisely to the Euclidean norm; and | · | is the Euclidean norm. Let A ≡ {x | f(x) = 0}, the 0-level set of f, be such that A 6= ∅. Observe that Theorem 2.2.4 implies that A is an (n−1)-dimensional submanifold, which implies that Hn−1(A) > 0. (In fact, A is an embedded submanifold without boundary.) Since A is closed, we have that A is compact

(as a subset of Rn). Further, by Theorem 2.2.1, A is (locally) the graph of a C1,1 function and thus has positive reach, which intuitively says that the normals to A do not intersect in an neighborhood of A for all < reach(A). Now, recall the Tube Formula [54] for the

n H -dimensional volume of the -“tube” T surrounding A:

n n−1 H (T) = 2H (A) + o(). (3.2)

n n−1 Thus T ⊂ B(0,R +) ⇒ H (T) < ∞, which together with (3.2) implies that H (A) < ∞ as well. Let > 0 and L be the non-negative Lipschitz constant of Df (see (3.5) below) such that L ||Dzf|| for all z ∈ A (which we can do since ||Df|| > 0 on A). Further, let ||Df|| > 0 in the -neighborhood E of A, and suppose the normals to A do not intersect in E (due to the positive reach of A). Next observe that we only need to concern ourselves with

22 1 integrating (3.1) over E (where ||Df||= 6 0!) since the k in the limit sends everything outside of that neighborhood to 0. This follows from the fact that the integrand is continuous on the compact set B(0, 2R) \ E. The idea that follows is that we want to integrate over E so as to integrate ﬁrst along the normals to A and then integrate along A, yet due to the expansion (contraction) that occurs in dimensions n ≥ 2 as we push our 0-level set A outward (inward) to form the neighborhood E, which is captured in the Tube Formula (3.2), we cannot simply apply Fubini’s Theorem to (3.1). Instead, we use the second version (2.1) of the Area Formula to account for this expansion in the following way.

Let C (which we will think of as an n-dimensional manifold) be the “cylinder” formed from “lifting” A up by in the R direction of the product space Rn × R. We wish to apply

Fubini’s Theorem over C (which we cannot do over E due to the curvature of A.) So, we construct a diﬀeomorphism H(w): C → E (quasi) explicitly using the normal map n(z) of A, where z ∈ A. Using H(w) with the Area Formula, we then need to account for JH, the Jacobian of H, that acts as the “expansion factor” of the transformation H(w); i.e. the amount of extra n-volume we get by expanding outward along the normals to A. But this we do by showing that as → 0,JH → 1. In our case, the Area Formula is given by

Z Z g(w) JH dHnw = g(H−1(y))dy, (3.3) C H(C)

n where w = (z, t) ∈ C ⊂ R × R, and t ∈ [−, ]. Now, setting k−1 1 ||Df(y)|| k g(H−1(y)) = k |f(y)| and H(z, t) = ι(z) + t n(ι(z)), where ι(z): Rn−1 → Rn is the inclusion map, we let ι(z) = z below, and we get that

23 k−1 Z Z 1 ||Df(z + t n(z))|| k g(w) JH dHnw = JH dHnw. (3.4) C C k |f(z + t n(z))|

We can now use Fubini’s on this, integrating, within C, ﬁrst in the t direction and then in the z direction (see Figure 3.1).

H(w) f −1{0}

n R

C R

f −1{0} Rn

Figure 3.1: A diﬀeomorphism H : C → E.

Now, with z ∈ A, y = z + t n(z) ∈ E, and t ∈ [−, ] as above, we seek a bound on

||Df(z + t n(z))|| = ||Dy|| as well as f(y) in our neighborhood E. Here, we exploit the Lipschitz condition of Df, letting 0 ≤ L ≡ the Lipschitz constant of Df. We recall that f ∈ C1,1 implies that its derivative is Lipschitz, so we have, for all x, v ∈ Rn,

||Dvf − Dxf|| ≤ L|v − x|. (3.5)

From this, we ﬁnd that

24 | ||Dyf|| − ||Dzf|| | ≤ ||Dyf − Dzf|| ≤ L|y − z| ≤ L,

which says

||Dzf|| − L ≤ ||Dyf|| = ||Df(z + t n(z))|| ≤ ||Dzf|| + L. (3.6)

It follows that

(||Dzf|| − L)|t| ≤ |f(z + t n(z))| ≤ (||Dzf|| + L)|t|,

and thus

||Dzf|| − L ||Df(z + t n(z))|| ||Dzf|| + L ≤ ≤ , L ||Dzf||. (3.7) (||Dzf|| + L)|t| |f(z + t n(z))| (||Dzf|| − L)|t|

We note that there is a more geometric way to obtain the bound on ||Dyf|| in (3.6)

h by looking at the ratio t , where h is the increase (decrease) in “rise” that f can obtain over the distance t (so here we assume t > 0) compared to the “rise” that Dzf obtains over t, where t is the length along the normal from z. This approach yields the same

∗ bounds as above (see Figure 3.2). To see this, we let ||Dyf∗|| and ||Dyf || represent the minimum and maximum values ||Df|| takes on E, respectively, and we look at the case

where 0 < ||Df∗|| ≤ ||Dyf|| ≤ ||Dzf||; the right hand side of (3.6) holds in a similar way if

∗ ||Dzf|| ≤ ||Dyf|| ≤ ||Df || < ∞.

Notice that in Figure 3.2, we see that h < ||Dzf||t − ||Df∗||t. The right side provides

us with the greatest diﬀerence possible by assuming that ||Df|| takes the value ||Df∗|| at each point along the normal to z between and including z and y. This leads to the following inequality. h ≤ | ||D f|| − ||D f || | ≤ ||D f − D f || ≤ L|z − y| ≤ L t z y ∗ z y ∗

25 And so, h ||D f|| − L ≤ ||D f|| − = ||D f||. z z t y

h Dz f Dy f

Df∗ y z

Figure 3.2: Bounding the derivative at y.

Since A has positive reach, we know from Section 5.8 that

(1 − κˆ)n−1 ≤ JH ≤ (1 + κˆ)n−1, (3.8)

1 whereκ ˆ ≡ reach(A) .

Now, putting this altogether, we have

26 ( k−1 ) 1 Z ||Df|| k lim dy (3.9) k→∞ k B(0,2R) |f| ( k−1 ) ( k−1 ) 1 Z ||Df|| k 1 Z ||Df|| k = lim dy + lim dy k→∞ k B(0,2R)\E |f| k→∞ k E |f| ( k−1 ) 1 Z ||Df(y)|| k = lim dy k→∞ k E |f(y)| ( k−1 ) 1 Z ||Df(z + t n(z))|| k = lim JHdHnw k→∞ k C |f(z + t n(z))| ( k−1 ) 1 Z Z ||Df(z + t n(z))|| k = lim JHdt dHn−1z k→∞ k A − |f(z + t n(z))| ( k−1 ) Z Z k 1 ||Dzf|| + L n−1 n−1 ≤ lim (1 + κˆ) dt dH z L ||Dzf|| k→∞ k A − (||Dzf|| − L)|t| ( k−1 k−1 ) 1 Z ||D f|| + L k Z 1 k = (1 + κˆ)n−1 lim z dt dHn−1z k→∞ k A ||Dzf|| − L − |t| ( k−1 ) Z k n−1 1 ||Dzf|| + L n−1 = 2(1 + κˆ) lim t k dH z (3.10) k→∞ 0 A ||Dzf|| − L

k−1 k Since, by the Dominated Convergence Theorem with g = 2 ≥ ||Dzf||+L , ||Dzf||−L

( k−1 ) Z ||D f|| + L k Z ||D f|| + L lim z dHn−1z = z dHn−1z, k→∞ A ||Dzf|| − L A ||Dzf|| − L we split the limit in (3.10) to give the following:

( k−1 ) Z k n−1 1 ||Dzf|| + L n−1 2(1 + κˆ) lim t k dH z (3.10) k→∞ 0 A ||Dzf|| − L ( k−1 ) Z k n−1 n 1 o ||Dzf|| + L n−1 = 2(1 + κˆ) lim k lim dH z k→∞ k→∞ A ||Dzf|| − L Z ||D f|| + L = 2(1 + κˆ)n−1 z dHn−1z. (3.11) A ||Dzf|| − L

27 We have thus far shown that

( k−1 ) 1 Z ||Df|| k Z ||D f|| + L lim dy ≤ 2(1 + κˆ)n−1 z dHn−1z. (3.12) k→∞ k B(0,2R) |f| A ||Dzf|| − L

But ||Dzf||+L → 1 uniformly as → 0 since setting = 1 and picking η > 0, we ﬁnd ||Dzf||−L n n

that ||Dzf||+L/n − 1 = 2L < η for all z ∈ A for n large enough. Letting → 0 in ||Dzf||−L/n n||Dzf||−L (3.12), we have

( k−1 ) 1 Z ||Df|| k lim dy k→∞ k B(0,2R) |f| Z ||D f|| + L ≤ 2 lim (1 + κˆ)n−1 z dHn−1z →0 A ||Dzf|| − L Z ||D f|| + L = 2 lim (1 + κˆ)n−1 lim z dHn−1z →0 →0 A ||Dzf|| − L Z ||D f|| + L = 2 lim z dHn−1z A →0 ||Dzf|| − L Z = 2 dHn−1 A = 2Hn−1(A). (3.13)

An analogous argument with the left inequalities of (3.7) and (3.8) above yields the reverse inequality to (3.13), and we have

( k−1 ) 1 Z ||Df|| k lim dx = 2Hn−1({x | f(x) = 0}). k→∞ k B(0,2R) |f|

Remark 3.1.1. Under the following assumptions, we can measure any level set for f in

Theorem 3.1. If c ∈ f(Rn) is a regular value and f −1{c} ⊂ B(0,R), then we simply deﬁne g ≡ f − c and use g in place of f in the theorem.

28 3.2 A Simple Example

Let f : R2 → R be f = x2+y2−1, a paraboloid with 0-level set E ≡ {(x, y) | x2+y2 = 1}, i.e. the unit circle. We know that twice the length of this set is 4π, so we will show that the integral above indeed yields this result (see Figure 3.3).

Figure 3.3: The graph of f = x2 + y2 − 1 with 0-level set the unit circle.

For functions mapping from Rn → R, ||Df|| is simply the vector norm of the gradient since the operator norm measures the greatest expansion of any unit vector that occurs under Df, which occurs exactly in the direction of the gradient. Thus, ∇f = (2x, 2y), and |∇f| = 2px2 + y2. If we consider B(0, 1) to be closed, then we shall set B(0, 2R) = B(0, 2).

29  k−1  ( k−1 ) p ! k 1 Z ||Df|| k 1 Z x2 + y2  lim d~x = 2 lim 2 2 dxdy k→∞ k B(0,2R) |f| k→∞ k B(0,2) x + y − 1   k−1  ! k 1 Z px2 + y2  = 2 lim 2 2 dxdy k→∞ k B(0,1) 1 − x − y   k−1  ! k 1 Z px2 + y2  + 2 lim 2 2 dxdy k→∞ k B(0,2)\B(0,1) x + y − 1  ( k−1 ) 1 Z 2π Z 1 r k = 2 lim 2 rdrdθ k→∞ k 0 0 1 − r ( k−1 ) 1 Z 2π Z 2 r k + 2 lim 2 rdrdθ (3.14) k→∞ k 0 1 r − 1

Using numerical integration techniques, we ﬁnd the value of (3.14) rapidly converging to 4π ≈ 12.566 as predicted. For example, with k = 1000, we have

( 999 ) 1 Z 2π Z 1 r 1000 2 lim 2 rdrdθ k→∞ 1000 0 0 1 − r ( 999 ) 1 Z 2π Z 2 r 1000 +2 lim 2 rdrdθ k→∞ 1000 0 1 r − 1

≈ 12.568. (3.15)

30 For an analytic solution, Mathematica R provides the following solution to (3.14).

( k−1 ) ( k−1 ) 1 Z 2π Z 1 r k 1 Z 2π Z 2 r k 2 lim 2 rdrdθ + 2 lim 2 rdrdθ k→∞ k 0 0 1 − r k→∞ k 0 1 r − 1 ( −1 3 1 1 ) 1 2 k Γ − Γ = 2π lim 2 2k k k→∞ k 1 1 Γ 2 3 + k ( " −1 1 #) 2 k Γ 1 k + 1 k ˜ k − 1 k + 1 k − 1 1 + 2π lim Γ − − 2 2F1 , − ; ; k→∞ k 2k 1 1 k 2k 2k 4 Γ 2 k − 1 1 −1 1 = 2π lim 2 k Γ k→∞ k k 1 −1 1 −1 ˜ −1 1 1 + 2π lim 2 k Γ − 2Γ 2F1 1, ; ; k→∞ k k 2 2 2 4 2 −1 −2 + coth−1(2) = 2π lim 1 + 1 − Γ √ k→∞ k 2 2 π 2 √ = 2π lim 1 + 1 − (−2 π)(.409 ... ) k→∞ k = 4π (3.16)

We note that the regularized hypergeometric function

F (a, b; c; z) F˜ (b, a; c; z) ≡ 2 1 √ , 2 1 π

where 2F1(a, b; c; z) is the hypergeometric function, which in turn is deﬁned by the power series ∞ n X (a)n(b)n z . (c) n! n=0 n

Here |z| < 1 and (q)n is the rising Pochhammer symbol deﬁned by

  1 n = 0 (q)n =  q(q + 1) ··· (q + n − 1) n > 0.

1 1 Further, note above that limk→∞ k Γ k = 1. We next present the second main result, that of computing the H1-measure of an em-

31 bedded C1 submanifold in R2 using the singular integral in (3.1) above.

32 CHAPTER 4

A C1-BOUNDARY MEASURE

4.1 The n = 2 Case

Theorem 4.1.1. Let f(x) = inf{|x − y| | y ∈ A} : R2 → R be the distance function to A ⊂ R2. Further, suppose A is a C1 closed embedded 1-dimensional submanifold of R2 such that A ⊂ B(0,R) (closed) and H1(A) < ∞. Then we have

( k−1 ) 1 Z |Df| k lim dy = 2H1(A). (4.1) k→∞ k B(0,2R) |f|

Proof. As A is a closed submanifold, i.e. it is compact without boundary, we assume A 6= ∅, which implies that H1(A) > 0. We also note that |Df| ≡ 1 H2-a.e.; Df is undeﬁned on A itself; and that A being C1 means that A is locally the graph of a C1 function g : R → R.

1. (a) Since A is an embedded submanifold, then globally speaking, no matter how close one portion of A comes to another portion, there is always at least a distance η∗ > 0 between them. We will chose η below in such a way that η ≤ η∗/4 so that we may cover A with rectangles extending outward η on both sides without overlapping (see Figure 4.1).

η ≡ η∗/4 η∗

Figure 4.1: Global distance of A to itself.

33 (b) Since A is a compact submanifold of R2, it is a compact set in R2, hence we

can cover A with a ﬁnite number of closed disks B(xi, δ) with xi ∈ A for i = 1, 2,...,M(δ), where δ is the ﬁneness of our cover and M(δ) is the number of

1 disks. Since A is C , for each xi, we deﬁne Ti ≡ Txi A∩B(xi, δ), the portion of the

linear approximation to A at the point xi contained in each ball B(xi, δ), which

is naturally a diameter of B(xi, δ) and thus has length 2δ (see Figure 4.2).

B(xi, δ)

Figure 4.2: Covering A with disks B(xi, δ).

(c) Using the fact that f is continuous on the compact set B(0, 2R) and that |Df| ≡ 1 H2-a.e., we see that the integrand in (4.1) is bounded outside of a neighborhood E of A contained in B(0, 2R) (see Figure 4.3). Thus we have

( k−1 ) ( k−1 ) 1 Z |Df| k 1 Z |Df| k lim dy = lim dy . (4.2) k→∞ k B(0,2R) |f| k→∞ k E |f|

34 B(0, 2R)

A E

Figure 4.3: The 1/k sends everything outside E to 0 in the limit.

η 2. (a) Let Ei be a rectangle that is centered on Ti with height 2η and width 2δ for i = 1, 2,...,M(δ) and with η chosen as above. We will integrate in (4.1) over each η ~ Ei and sum the results so that we can use the normals N to each Ti to approximate η the normals ~n to A ∩ Ei . However, we need to bound how much “overlap” there η η M(δ) is between the Ei ’s so that the integral over the collection {Ei }i=1 converges to M(δ) η the integral over ∪i=1 Ei as δ → 0. We do this in the following way, noting that M(δ) η since A ⊂ ∪i=1 Ei ⊂ B(0, 2R), we can apply (4.2) above (see Figure 4.4).

35 Ti

xi η Ei

η Figure 4.4: Covering A with rectangles Ei .

(b) Let w, z ∈ A and α > 0. By the uniform continuity of Dg, (locally) we can choose δ small enough so that

|w − z| < δ ⇒ |Dwg − Dzg| < α. (4.3)

But this implies that |g(w) − g(z)| < αδ ≡ . (4.4)

√ We pick δ small enough so that letting η ≡ αδ, we still have η ≤ η∗/4 from above. Then

√ η αδ 1 = = √ → ∞ (4.5) αδ α

as α, δ → 0.

Similarly,

δ δ 1 = √ = √ → ∞ (4.6) η αδ α

as α, δ → 0.

36 Lastly, we have δ δ 1 = = → ∞ (4.7) αδ α

as α, δ → 0.

η By the geometric deﬁnition of the derivative [55], we observe that A ∩ Ei is

contained in a cone Ki centered on xi and Ti of max height 2 (at a distance δ

from xi along Ti) and angular width 2θ (see Figure 4.5).

η θ Ki xi Ti δ A

η Ei

η Figure 4.5: For each xi,A ∩ Ei is contained the cone Ki . Note that the image is not drawn to scale since δ η .

η 3. (a) We now specify the manner in which we cover A with disks, and thus with Ei ’s. η We wish for the “overlaps” of the Ei ’s to contain at least an η/2-neighborhood of A, and we want

2 η η 2 2 H (Ei ∩ Ej ) ≤ (2η) = 4η , i 6= j.

η By choosing δ small enough, we can arrange each Ei so that it intersects with η its neighbors only in the 2η-squares on each end of each rectangle Ei (see Figure

37 4.6).

More precisely,

2 η 2 2 η η H (Ei ) = 2δ2η 4η ≥ H (Ei ∩ Ej ), i 6= j, (4.8)

since (4.6) above implies δ η. (4.9)

η Thus sending δ to 0 will send the total area of the pairwise “overlap” of the Ei ’s η M(δ) to 0 faster than the area of the collection {Ei }i=1 goes to 0.

2η

η Ei

η Figure 4.6: Overlaying A with Ei ’s

(b) We set k−1 1 Z |Df| k I = dz, k Ω |f|

η η η where Ω ≡ {Ei ∩Ej }i6=j, the collection of “overlap” of all the Ei ’s taken pairwise. Now we can write

38 ( k−1 ) 1 Z |Df| k lim dz k→∞ k M(δ) η |f| ∪i=1 Ei ( k−1 ) 1 Z |Df| k = lim dz − I k→∞ k η M(δ) |f| {Ei }i=1 ( k−1 ) 1 − λ Z |Df| k = lim dz k→∞ k η M(δ) |f| {Ei }i=1 (4.10)

since I is some fraction 0 ≤ λ ≡ λ(δ) < 1 of the same integral taken over the

η M(δ) whole (ﬁnite) collection {Ei }i=1 . Based on the argument just given, λ → 0 as δ → 0.

η 4. (a) We now wish to ﬁnd bounds on f by showing that the normals to A∩Ei converge

to the normals to Ti as δ → 0. The key lemma here is as follows:

∗ Lemma 4.1.1. Let > > 0 and let Ki be the cone of angular width 2θ centered

η ∗ on x ∈ Ti containing A ∩ Ei . There exists a cone Ci such that we may translate

η ∗ it (without tipping) along A ∩ Ei so that A is contained inside Ci for all points of A within δ∗ < δ of its center point (see Figure 4.7).

Proof. Let δ > 0 be as above and w, z ∈ A∩Ei . Then from the uniform continuity

of Dg, we ﬁnd that |w − z| < δ implies ∠(Dwg, Dzg) ≤ 2θ, that is the angle between Dwg and Dzg is bounded by 2θ, which deﬁnes the angular width of Ki . ~ Thus the normals ~n to A ∩ Ei can only deviate from the normals N to Ti by at

−1 most θ = tan ( δ ). ~ Now, let a ∈ A ∩ Ei and let N(x) be the normal to Ti containing a. Pick a point z = (x, t) ∈ N~ (x), where t ≤ and draw the line segments through z on each side of N~ (x) forming an angle θ with N~ (x). Without loss of generality, pick one of

these segments and label its intersection with A asa ˆ ∈ A ∩ Ei . We ﬁnd, from the

39 above argument, that ~n(ˆa), the normal toa ˆ, cannot deviate from its corresponding ~ N(ˆx) by more than θ, and thus ∠(Dag, Daˆg) ≤ 2θ. But this implies thata ˆ must ∗ be inside the cone Ci centered on a whose center line l is parallel to Ti.

Letting z = (x, t) be any point along N~ (x) with t ≤ , we ﬁnd that the corre-

∗ spondinga ˆ ∈ Ci , which proves the assertion. We observe that from (4.7), as δ → 0, θ also goes to 0.

η (b) As shown in Figure 4.7, using our lemma we wish to bound f(z) for z = (x, t) ∈ Ei

such that x ∈ Ti and t ∈ [−η, η] is the distance to x in the direction of the normal ~ ∗ η N(x) to Ti. But if we translate Ci to each z ∈ A ∩ Ei (keeping its center line l

parallel to Ti), we ﬁnd (with 0 ≤ θ < π/2)

cos(θ)|t∗| ≤ |f(z∗)| ≤ |t∗|, (4.11)

which implies

1 1 1 ≤ ≤ . (4.12) |t∗| |f(z∗)| cos(θ)|t∗|

Here, z∗ = (x, t∗), where t∗ represents the distance from z = (x, t) to a ∈ A along ~ the normal N(x) instead of the distance to x ∈ Ti. The actual distance t to Ti is

∗ ∗ either t + β or t − β, where 0 ≤ β ≤ , depending on what side of Ti a is and

since A is contained in Ki .

40 z = (x, t)

t∗ cos(θ)t∗

2θ θ C∗ a i l A Ti β x xi δ

∗ ∗ Figure 4.7: Translating Ci along A so that line l is always parallel to Ti and t is the distance from z to a.

Now, we wish to integrate 1/|f(z)| over [−η, η], with η being measured from Ti,

∗ but to use our bound in (4.12), we make the substitution z = (x, t) ± (0, βx),

∗ ∗ ∗ with t = t ± βx, so that dz = dz and dt = dt. Note that the points (x, t) and

(x, t∗) represent the same point in R2, but when integrating, we will have diﬀerent bounds in the integral since our datum is diﬀerent if our point a ∈ A is not also in

Ti, that is if a 6= x. Therefore, in the case when a is above Ti, we let our bounds

+ − beη ˆx = η − βx andη ˆx = −(η + βx), where βx depends on x ∈ Ti. In the case

+ − when a is below Ti, we haveη ˆx = η + βx andη ˆx = −(η − βx) (see Figure 4.8).

41 + η ηˆx = η − βx

a a l βx Ti

− −η ηˆx = −(η + βx) ∗ t=distance from Ti t = distance from l

Figure 4.8: On the left, we integrate 1/|f(z)| = 1/|f(x, t)| over [−η, η] with t measured as ∗ ∗ − + the distance from x ∈ Ti. On the right, we integrate 1/|f(z )| = 1/|f(x, t )| over [ˆηx , ηˆx ] ∗ with t measured as the distance from a ∈ l. Both are for the case where a is above Ti.

In either case, we are integrating over the same interval. Since all terms in (4.12) are positive, this will be greater than integrating over t ∈ [−(η − ), η − ] and less than integrating over t ∈ [−(η + ), η + ], and we have

k−1 + k−1 Z η 1 k Z ηˆx 1 k dz = dz∗ − ∗ −η |f(z)| ηˆx |f(z )| + k−1 Z ηˆx 1 k ≤ dt∗ − ∗ ηˆx cos(θ)|t | k−1 Z η+ 1 k ≤ dt, (4.13) −(η+) cos(θ)|t|

where k = 1, 2, 3,... . Similarly,

42 k−1 + k−1 Z η 1 k Z ηˆx 1 k dz = dz∗ − ∗ −η |f(z)| ηˆx |f(z )| + k−1 Z ηˆx 1 k ≥ dt∗ − ∗ ηˆx |t | k−1 Z η− 1 k ≥ dt. (4.14) −(η−) |t|

5. Putting this all together, we now have that

( k−1 ) 1 Z |Df| k lim dz k→∞ k B(0,2R) |f| ( k−1 ) 1 Z |Df| k = lim dz k→∞ k M(δ) η |f| ∪i=1 Ei ( k−1 ) 1 Z |Df| k = lim dz − I k→∞ k η M(δ) |f| {Ei }i=1  M(δ) k−1  1 − λ X Z 1 k  = lim dz k→∞ k η |f|  i=1 Ei 

 M(δ) k−1  1 − λ X Z Z η 1 k  = lim dt dH1(x) k→∞ k |f|  i=1 Ti −η 

 M(δ) k−1  1 − λ X Z Z η+ 1 k  ≤ lim dt dH1(x) k→∞ k cos(θ)|t|  i=1 Ti −(η+) 

 k−1 k−1 M(δ)  1 − λ Z η+ 1 k 1 k X Z  = lim dt dH1(x) k→∞ k |t| cos(θ)  −(η+) i=1 Ti 

( k−1 )  k−1 M(δ)  Z η+ k k 1 − λ 1  1 X 1  = lim dt lim H (Ti) k→∞ k |t| k→∞ cos(θ) −(η+)  i=1  M(δ) n 1 o 1 X 1 = 2(1 − λ) lim (η + ) k H (Ti) k→∞ cos(θ) i=1 M(δ) 2(1 − λ) X = H1(T ). (4.15) cos(θ) i i=1

43 6. We now let δ → 0 in (4.15), since the above process holds for each cover of A we choose according to our procedure. Let δ = 1/n and deﬁne the partial sum of the series in (4.15) above as M(1/n) X 1 Sn ≡ H (Ti). i=1

1 We see that {Sn} is monotonically increasing and bounded by H (A) < ∞ except for

η an error term. This is due to the fact that in the “overlap” of the Ei ’s, there may be

normals to A that intersect more than one Ti with the result that a “double-counted”

length of at most 2η will be included in the sum above for each Ti. For a given δ, this error term is bounded by 2M(δ)η. Yet we know that η goes to 0 faster than δ goes to 0 (thus while M(δ) < ∞). Therefore, this error term vanishes as δ gets small enough,

1 and we have that {Sn} → H (A).

Letting n → ∞ above amounts to letting δ → 0, which forces θ, λ → 0 by our construction above. This gives

 M(δ)  2(1 − λ) X 1  1 lim H (Ti) = 2H (A). (4.16) δ→0 cos(θ)  i=1 

Thus we have shown that

( k−1 ) 1 Z |Df| k lim dx ≤ 2H1(A). k→∞ k B(0,2R) |f|

7. Using an analogous reasoning with the bound in (4.14), we obtain the desired result.

44 CHAPTER 5

CUBICAL COVERS OF SETS IN Rn

5.1 Representing Sets & their Boundaries in Rn

We now explain and illuminate a few ideas for (1) representing sets and (2) learning from those representations. Though some of the ideas and results we explain are likely written down elsewhere (though we are not aware of those references), our purpose is not to claim priority to those pieces, but rather to stimulate thought and exploration. Our primary intended audience is students of mathematics even though other, more mature mathematicians may ﬁnd a few of the ideas interesting. We believe that cubical covers can be used at an earlier point in the student career and that both the β numbers idea introduced by Peter Jones and the idea of varifolds pioneered by Almgren and Allard and now actively being developed by Menne, Buet, and collaborators are still very much underutilized by all (young and old!). To that end, we have written this exploration, hoping that the questions and ideas presented here, some rather elementary, will stimulate others to explore the ideas for themselves. We begin by brieﬂy introducing cubical covers, Jones’ β, and varifolds, after which we look more closely at questions involving cubical covers. Then both of the other approaches are explained in a little bit of detail, mostly as an invitation to more exploration, after which we close with problems for the reader and some unexplored questions.

5.2 Cubical Reﬁnements: Dyadic Cubes

In order to characterize various sets in Rn, we explore the use of cubical covers whose 1 cubes have side lengths which are positive integer powers of 2 , dyadic cubes, or more precisely, (closed) dyadic n-cubes with sides parallel to the axes. Thus the side length at the

1 dth subdivision is l(C) = 2d , which can be made as small as desired.

45 Figure 5.1 illustrates this by looking at a unit cube in R2 lying in the ﬁrst quadrant with a vertex at the origin. We then form a sequence of reﬁnements by dividing each side length in half successively, and thus quadrupling the number of cubes each time.

Deﬁnition 5.2.1. We shall say that the n-cube C (with side length denoted as l(C)) is dyadic if n Y −d −d C = [mj2 , (mj + 1)2 ], mj ∈ Z, d ∈ N ∪ {0}. j=1

(0, 1) (1, 1)

(0, 0) (1, 0) d = 0 d = 1 d = 2 l(C) = 1 l(C) = 1 1 20 21 l(C) = 22

Figure 5.1: Dyadic cubes.

In this paper, we will assume C to be a dyadic n-cube throughout. We will denote the

1 n E union of the dyadic n-cubes with edge length 2d that intersect a set E ⊂ R by Cd and E deﬁne ∂Cd to be the boundary of this union (see Figure 5.2). Two simple questions we will explore for their illustrative purposes are:

n E n 1. “If we know L (Cd ), what can we say about L (E)?” and similarly,

n−1 E n−1 2. “If we know H (∂Cd ), what can we say about H (∂E)?”

46 E

E Cd

E Figure 5.2: Cubical cover Cd of a set E.

5.3 Jones’ β Numbers

Another approach to representing sets in Rn, developed by Jones [29], and generalized by Okikiolu [49], Lerman [33], and Schul [51], involves the question of under what conditions a bounded set E can be contained within a rectifiable curve Γ, which Jones likened to the Traveling Salesman Problem taken over an infinite set. (See Definition 5.7.2 below for the definition of rectifiable.) Jones showed that if the aspect ratios of the optimal containing cylinders in each dyadic cube go to zero fast enough, the set E is contained in a rectifiable curve. Jones’ approach ends up providing one useful approach of defining a representation for a set in Rn similar to those discussed in the next section. We return to this topic in Section 5.10. The basic idea is illustrated in Figure 5.3.

47 C

Figure 5.3: Jones’ β numbers. The green lines indicate the thinnest cylinder containing Γ in the cube C. We see from this relatively large width that Γ is not very “ﬂat” in this cube.

5.4 Working Upstairs: Varifolds

A third way of representing sets in Rn uses varifolds. Instead of representing E ⊂ Rn by working in Rn, we work in the Grassmann Bundle, Rn × G(n, m). We parameterize the Grassmannian G(2, 1) by taking the upper unit semicircle in R2 (including the point (1, 0), but not including (1, π), where both points are given in polar coordinates) and straightening it out into a vertical axis (as in Figure 5.4). The bundle

R2 × G(2, 1) is then represented by R2 × [0, π).

48 π

π 0

Figure 5.4: The vertical axis for the “upstairs.”

π π S1 S2

π S π 2 2

2 2 0 R 0 R E2 E1 E

Figure 5.5: Working upstairs in the Grassmann bundle.

Figure 5.5 illustrates how the tangents are built into this representation of subsets of

Rn, giving us a sense of why this representation might be useful. A circular curve in R2 becomes two half-spirals upstairs (in the Grassmann bundle representation, as shown in the

ﬁrst image of Figure 5.5). Other curves in R2 are similarly illuminated by their Grassmann bundle representations. We return to this idea in Section 5.11.

5.5 Simple Questions

Let E ⊂ Rn and C be any dyadic n-cube as before. Deﬁne

C(E, d) = {C | C ∩ E 6= ∅, l(C) = 1/2d}

49 and, as above,

E [ Cd ≡ C. C∈C(E,d) Here are two questions:

n 1. Given E ⊂ R , when is there a d0 such that for all d ≥ d0, we have

n E n L (Cd ) ≤ M(n)L (E) (5.1)

for some constant M(n) independent of E?

n 2. Given E ⊂ R , and any δ > 0, when does there exists a d0 such that for all d ≥ d0, we have

n E n L (Cd ) ≤ (1 + δ)L (E)? (5.2)

Remark 5.5.1. Of course using the fact that Lebesgue measure is a Radon measure, we can very quickly get that for d large enough (i.e. 2−d small enough), the measure of the cubical cover is as close to the measure of the set as you want, as long as the set is compact and has positive measure. But the focus of this paper is on what we can get in a much more transparent, barehanded fashion, so we explore along diﬀerent paths, getting answers that are, by some metrics, suboptimal.

n n n n E Example 5.5.1. If E = Q ∩ [0, 1] , then L (E) = 0, but L (Cd ) = 1 ∀d ≥ 0.

Example 5.5.2. Let E be as in Example 5.5.1. Enumerate E as qˆ1, qˆ2, qˆ3,.... Now let

n n 1 Di = B(qˆi, 2i ) and E ≡ {∪Di} ∩ [0, 1] with chosen small enough so that L (E) ≤ 100 .

n 1 n E Then L (E) ≤ 100 , but L (Cd ) = 1 ∀d > 0.

5.6 A Union of Balls

n ¯ For a given set F ⊆ R , suppose E = ∪x∈F B(x, r), a union of closed balls of radius r centered at each point x in F . Then we know that E is regular (locally Ahlfors n-regular

50 or locally n-regular), and thus there exist 0 < m < M < ∞ and an r0 > 0 such that for all x ∈ E and for all 0 < r < r0, we have

mrn ≤ Ln(B¯(x, r) ∩ E) ≤ Mrn.

This is all we need to establish a suﬃcient condition for Equation (5.1) above.

Remark 5.6.1. The upper bound constant M is immediate since E is a union of n-balls, so

M = αn, the n-volume of the unit n-ball, works. However, this is not the case for k-regular sets in Rn, k < n, since we are now asking for a bound on the k-dimensional measure of an n-dimensional set which could easily be inﬁnite.

¯ 1. Suppose E = ∪x∈F B(x, r), a union of closed balls of radius r centered at each point x in F .

1 ˆ 2. Let C = C(E, d) for some d such that 2d r, and let C = {3C | C ∈ C}, where 3C is an n-cube concentric with C with sides parallel to the axes and l(3C) = 3l(C), as shown in Figure 5.6.

Figure 5.6: Concentric cubes.

3. This implies that for 3C ∈ Cˆ

Ln(3C ∩ E) > θ > 0, with θ ∈ . (5.3) Ln(3C) R

51 4. We then make the following observations:

(a) Note that there are 3n diﬀerent tilings of the plane by 3C cubes whose vertices

1 n live on the 2d lattice. (This can be seen by realizing that there are 3 shifts you can perform on a 3C cube and both (1) keep the originally central cube C in the

1 3C cube and (2) keep the vertices of the 3C cube in the 2d lattice.)

n (b) Denote the 3C cubes in these tilings Ti, i = 1, ..., 3 .

ˆ ˆ (c) Deﬁne Ci ≡ C ∩ Ti.

ˆ (d) Note now that by Step (3), the number of 3C cubes in Ci cannot exceed

Ln(E) N ≡ . i θLn(3C)

(e) Denote the total number of cubes in C by N E . Cd

(f) The number of cubes in C, N E , cannot exceed Cd

3n X Ln(E) N = 3n . i θLn(3C) i=1

(g) Putting it all together, we get

n E n L (Cd ) = L (∪C∈CC)

n = N E L (C) Cd Ln(E) ≤ 3n Ln(C) θLn(3C) Ln(E) = . (5.4) θ

¯ 5. This shows that if E = ∪x∈F B(x, r), then

1 Ln(CE) ≤ Ln(E). d θ

52 We now have two conclusions:

Regularized sets: We notice that for any ﬁxed r0 > 0, as long as we pick d0 big enough, ¯ then r < r0 and d > d0 imply that E = ∪x∈F B(x, r) satisﬁes

1 Ln(CE) ≤ Ln(E), d θ(n)

for a θ(n) > 0 that depends on n, but not on F .

Regular sets: Now suppose that

n n n ¯ F ∈ Rm ≡ {W ⊂ R | mr < L (W ∩ B(x, r)), ∀ x ∈ W and r < r0}.

Then we immediately get the same result: for a big enough d (depending only on r0),

1 Ln(CF ) ≤ Ln(F ), d θ(m)

where θ(m) > 0 depends only on the regularity class that F lives in and not on which subset in that class we cover with the cubes.

5.7 Minkowski Content

n Deﬁnition 5.7.1. Let W ⊂ R , and let Wr ≡ {x | d(x, W ) < r}. The (n − 1)-dimensional n n−1 L (Wr) Minkowski Content is deﬁned as M (W ) ≡ limr→0 2r , when the limit exists (see Figure 5.7).

Definition 5.7.2. A set W ⊂ Rn is called (Hm, m)-rectifiable if Hm(W ) < ∞ and Hm- almost all of W is contained in the union of the images of countably many Lipschitz functions from Rm to Rn. We will use rectifiable and (Hm, m)-rectifiable interchangeably when the dimension of the sets are clear from the context.

53 Wr W

Figure 5.7: Minkowski content.

Deﬁnition 5.7.3. We will say that E ⊂ Rn is m-rectiﬁable if there is a Lipschitz function mapping a bounded subset of Rm onto E.

Theorem 5.7.1. Mn−1(W ) = Hn−1(W ) when W is a closed, (n-1)-rectiﬁable set.

See Theorem 3.2.39 in [23] for a proof.

Remark 5.7.1. Notice that m-rectifiable is more restrictive that (Hm, m)-rectifiable. In fact, Theorem 5.7.1 is false for (Hm, m)-rectifiable sets. See the notes at the end of Section 3.2.39 in [23] for details.

√ 1 Now, let W be (n-1)-rectiﬁable, set rd ≡ n 2d , and choose rδ small enough so that

n n−1 L (Wrd ) ≤ M (W )2rd + δ,

for all d ∈ N ∪ {0} such that rd ≤ rδ. (Note: Because the diameter of an n-cube with edge 1 √ 1 W length 2d is rd = n 2d , no point of Cd can be farther than rd away from W . Thus W Cd ∈ Wrd .)

54 Assume that Ln(E) 6= 0 and ∂E is (n-1)-rectiﬁable. Letting W ≡ ∂E, we have

n E n n L (Cd ) − L (E) ≤ L (Wrd )

n−1 ≤ M (∂E)2rd + δ

n−1 ≤ M (∂E)2rδ + δ so that Mn−1(∂E)2r + δ Ln(CE) ≤ (1 + δˆ)Ln(E), where δˆ = δ . (5.5) d Ln(E) ˆ Since we control rδ and δ, we can make δ as small as we like, and we have a suﬃcient condition to establish Equation (5.2) above.

The result: let δˆ be as in Equation (5.5) and E ⊂ Rn such that Ln(E) 6= 0. Suppose that ∂E (which is automatically closed) is (n-1)-rectiﬁable and Hn−1(∂E) < ∞, then, for every

δ > 0 there exists a d0 such that for all d ≥ d0,

n E ˆ n L (Cd ) ≤ (1 + δ)L (E).

n Problem 5.7.1. Suppose that E ⊂ R is bounded. Show that for any r > 0, Er, the set of points that are at most a distance r from E, has a (Hn−1, n − 1)-rectiﬁable boundary. Show this by showing that ∂Er is contained in a ﬁnite number of graphs of Lipschitz functions from

n−1 R to R. Hint: cut Er into small chunks Fi with common diameter D r and prove that

(Fi)r is the union of a ﬁnite number of Lipschitz graphs.

Problem 5.7.2. Can you show that in fact the boundary of Er, ∂Er, is actually (n-1)- rectiﬁable? See if you can use the results of the previous problem to help you.

Remark 5.7.2. We can cover a union E of open balls of radius r, whose centers are bounded,

E E with a cover Cd satisfying Equation (5.2). In this case, ∂Cd certainly meets the requirements for the result just shown.

55 5.8 Smooth Boundary, Positive Reach

In this section, we show that if ∂E is smooth (at least C1,1), then E has positive reach allowing us to get an even cleaner bound, depending in a precise way on the curvature of ∂E.

n We will assume that E is closed. Deﬁne Er = {x ∈ R | dist(x, E) ≤ r}, clos(x) ≡ {y ∈ E | d(x, E) = |x − y|} and unique(E) = {x | clos(x) is a single point}.

Deﬁnition 5.8.1. The reach of E, reach(E), is deﬁned

reach(E) ≡ sup{r | Er ⊂ unique(E)}

Remark 5.8.1. Sets of positive reach were introduced by Federer in 1959 [22] in a paper that also introduced the famous coarea formula.

Remark 5.8.2. If E ⊂ Rn is (n-1)-dimensional and E is closed, then E = ∂E.

Another equivalent deﬁnition involves rolling balls around the boundary of E. The closed ball B¯(x, r) touches E if

B¯(x, r) ∩ E ⊂ ∂B¯(x, r) ∩ ∂E

Deﬁnition 5.8.2. The reach of E, reach(E), is deﬁned

reach(E) ≡ sup{r | every ball of radius r touching E touches at a single point}.

Put a little more informally, reach(E) is the supremum of radii r of the balls such that each ball of that radius rolling around E touches E at only one point (see Figure 5.8).

56 Positive Reach Non-positive Reach

Figure 5.8: Positive and non-positive reach.

As mentioned above, if ∂E is C1,1, then it has positive reach (see Remark 4.20 in [22]).

n That is, if for all x ∈ ∂E, there is a neighborhood of x, Ux ⊂ R , such that after a suitable

1,1 n−1 change of coordinates, there is a C function f : R → R such that ∂E ∩ Ux is the graph of f. (Recall that a function is C1,1 if its derivative is Lipschitz continuous.) This implies, among other things, that the (symmetric) second fundamental form of ∂E exists Hn−1-almost everywhere on ∂E. The fact that ∂E is C1,1 implies that at Hn−1-almost every point of ∂E,

1 the n − 1 principal curvatures κi of our set exist and |κi| ≤ reach(∂E) for 1 ≤ i ≤ n − 1. We will use this fact to determine a bound for the (n − 1)-dimensional change in area as the boundary of our set is expanded outwards or contracted inwards by (see Figure 5.9,

Diagram 1). Let us ﬁrst look at this in R2 by examining the following ratios of lengths of expanded or contracted arcs for sectors of a ball in R2 as shown in Diagram 2 in Figure 5.9 below.

l l

l−

Diagram 1. Diagram 2.

Figure 5.9: Moving out and sweeping in.

57 H1(l ) (r + )θ = = 1 + = 1 + κ H1(l) rθ r

H1(l ) (r − )θ − = = 1 − = 1 − κ, H1(l) rθ r

where κ is the principal curvature of the circle (the boundary of the 2-ball), which we can

think of as deﬁning the reach of a set E ⊂ R2 with C1,1-smooth boundary.

The Jacobian for the normal map pushing in or out by , which by the area formula is Qn−1 the factor by which the area changes, is given by i=1 (1 ± κi) (see Figure 5.9, Diagram 1).

If we deﬁneκ ˆ ≡ max{|κ1|, |κ2|,..., |κn−1|}, then we have the following ratios:

Max Fractional Increase of Hn−1 boundary “area” Moving Out:

n−1 Y n−1 (1 + κi) ≤ (1 + κˆ) . i=1

Max Fractional Decrease of Hn−1 boundary “area” Sweeping In:

n−1 Y n−1 (1 − κi) ≥ (1 − κˆ) . i=1

1 Remark 5.8.3. Notice that κˆ = reach(∂E) .

For a ball, we readily ﬁnd the value of the ratio

Ln(B(0, r + )) r + n = Ln(B(0, r)) r = (1 + κ)n (setting δ = κ)

= (1 + δ)n, (5.6)

1 where κ = r is the curvature of the ball along any geodesic.

58 Now we calculate the bound we are interested in for E, assuming ∂E is C1,1. Deﬁne

n E ⊂ R ≡ {x | d(x, E) < }. We ﬁrst compute a bound for

Ln(E ) Ln(E) + Ln(E \ E) = Ln(E) Ln(E) Ln(E \ E) = 1 + . (5.7) Ln(E)

n−1 Since κi is a function of x ∈ ∂E defined H -almost everywhere, we may set up the integral below over ∂E and do the actual computation over ∂E \ K, where K ≡ {the set of measure 0 where κi is not defined}. Computing bounds for the numerator and denominator separately in the second term in (5.7), we find, by way of the Area Formula [47],

Z Z n−1 n Y n−1 L (E \ E) = (1 + rκi)dH dr 0 ∂E i=1 Z Z ≤ (1 + rκˆ)n−1dHn−1dr 0 ∂E n n−1 (1 + rκˆ) = H (∂E) nκˆ 0 (1 + κˆ)n 1 = Hn−1(∂E) − (5.8) nκˆ nκˆ

and

n−1 Z r0 Z n Y n−1 L (E) ≥ (1 − rκi)dH dr 0 ∂E i=1 Z r0 Z ≥ (1 − rκˆ)n−1dHn−1dr 0 ∂E n r0 n−1 −(1 − rκˆ) = H (∂E) nκˆ 0 Hn−1(∂E) 1 = , when r = . (5.9) nκˆ 0 κˆ

59 From 5.7, 5.8, and 5.9, we have

n−1 (1+κˆ)n 1 n L (E ) H (∂E) nκˆ − nκˆ ≤ 1 + Ln(E) Hn−1(∂E) nκˆ = (1 + κˆ)n (setting δ = κˆ)

= (1 + δ)n. (5.10)

From this we get that

n n n L (E) ≤ (1 + κˆ) L (E) so that

n E n n L (Cd()) ≤ (1 + κˆ) L (E)

√ n √ 1 where d() = log2( ) is found by solving n 2d = . Thus, when ∂E is smooth enough to have positive reach, we ﬁnd a nice bound of the type in Equation (5.2), with a precisely known dependence on curvature.

5.9 A Boundary Conjecture

What can we say about boundaries? Can we bound

Hn−1(∂CE) d ? Hn−1(∂E)

E ∂Cd ∂E E

Figure 5.10: Cubes on the boundary.

60 Conjecture 5.9.1. If E ⊂ Rn is compact and ∂E is C1,1,

n−1 E H (∂Cd ) lim sup n−1 ≤ n. d→∞ H (∂E)

Brief Sketch of Proof for n = 2.

1. Since ∂E is C1,1, we can zoom in far enough at any point x ∈ ∂E so that it looks ﬂat.

2. Let C be a cube in the cover C(E, d) that intersects the boundary near x and has faces

E E in the boundary ∂Cd . Deﬁne F = ∂C ∩ ∂Cd .

3. (Case 1) Assume that the tangent at x, Tx∂E, is not parallel to either edge direction of the cubical cover (see Figure 5.11).

H1(F ) (a) Let Π be the projection onto the horizontal axis and notice that Π(F ) ≤ 2 + for any epsilon.

(b) This is stable to perturbations which is important since the actual piece of the boundary ∂E we are dealing with is not a straight line.

4. (Case 2) Suppose that the tangent at x, Tx∂E, is parallel to one of the two faces of

the cubical cover, and let Ux be a neighborhood of x ∈ ∂E.

(a) Zooming in far enough, we see that the cubical boundary can only oscillate up and down so that the maximum ratio for any horizontal tangent is (locally) 2.

(b) But we can create a sequence of examples that attain ratios as close to 2 as we like by ﬁnding a careful sequence of perturbations that attains a ratio locally of 2 − for any (see Figure 5.10).

∞ (c) That is, we can create perturbations that, on an unbounded set of d’s, {di}i=1, 1 E H (C ∩ Ux) di yield a ratio ∂E > 2 − , and we can send → 0.

5. Use the compactness of ∂E to put this all together into a complete proof.

61 E ∂Cd ∂E θ

Figure 5.11: The case in which θ, the angle between Tx∂E and the x-axis, is neither 0 nor π/2.

Problem 5.9.1. Suppose we exclude C’s that contain less than some fraction θ of E (as

ˆE deﬁned in Conjecture 5.9.1) from the cover to get the reduced cover Cd . In this case, what is the optimal bound B(θ) for the ratio of boundary measures

n−1 ˆE H (∂Cd ) lim sup n−1 ≤ B(θ)? d→∞ H (∂E)

5.10 The Jones’ β Approach

As mentioned above, another approach to representing sets in Rn, developed by Jones [29], and generalized by Okikiolu [49], Lerman [33], and Schul [51], involves the question of under what conditions a bounded set E can be contained within a rectiﬁable curve Γ, which Jones likened to the Traveling Salesman Problem taken over an inﬁnite set. While Jones

worked in C in his original paper, the work of Okikiolu, Lerman, and Schul extended the results to Rn ∀n ∈ N as well as inﬁnite dimensional space.

62 Recall that a compact, connected set Γ ⊂ R2 is rectiﬁable if it is contained in the image of a countable set of Lipschitz maps from R into R2, except perhaps for a set of H1 measure zero. We have the result that if Γ is compact and connected, then l(Γ) = H1(Γ) < ∞ implies it is rectiﬁable (see pages 34 and 35 of [21]).

Let WC denote the width of the thinnest cylinder containing the set E in the dyadic n-cube C (see Figure 5.12), and deﬁne the β number of E in C to be

W β (C) ≡ C . E l(C)

Figure 5.12: Jones’ β numbers and WC . Each of the two green lines in a cube C is an equal distance away from the red line and is chosen so that the green lines define the thinnest cylinder containing E ∩ C. Then the red lines are varied over all possible lines in C to find that red line whose corresponding cylinder is the thinnest of all containing cylinders. In this sense, the minimizing red lines are the best fit to E in each C.

Jones’ main result is this theorem:

Theorem 5.10.1. [29] Let E be a bounded set and Γ be a connected set both in R2. Deﬁne

WC βΓ(C) ≡ l(C) , where WC is the width of the thinnest cylinder in the 2-cube C containing Γ.

63 Then, summing over all possible C,

2 X 2 β (Γ) ≡ (βΓ(3C)) l(C) < η l(Γ) < ∞, where η ∈ R. C

Conversely, if β2(E) < ∞ there is a connected set Γ, with E ⊂ Γ, such that

2 l(Γ) ≤ (1 + δ) diam(E) + αδβ (E),

where δ > 0 and αδ = α(δ) ∈ R.

Jones’ main result, generalized to Rn, is that a bounded set E ⊂ Rn is contained in a rectiﬁable curve Γ if and only if

2 X 2 β (E) ≡ (βE(3C)) l(C) < ∞, C

where the sum is taken over all dyadic cubes. Note that each β number of E is calculated over the dyadic cube 3C, as defined in Section 5.6. Intuitively, we see that in order for E to lie within a rectifiable curve Γ, E must look flat as we zoom in on points of E since Γ has tangents at H1-almost every point x ∈ Γ. Since both WC and l(C) are in units of length, βE(C) is a scale-invariant measure of the flatness of E in C. In higher dimensions, the analogous cylinders’ widths and cube edge lengths are also divided to get a scale-invariant βE(C). The notion of local linear approximation has been explored by many researchers. See for example the work of Lerman and collaborators [16,6, 60,7]. While distances other than the sup norm have been considered when determining closeness to the approximating line, see [33], there is room for more exploration there. In the section below, Problems and Questions, we suggest an idea involving the multiscale flat norm from geometric measure theory.

64 5.11 A Varifold Approach

As mentioned above, a third way of representing sets in Rn uses varifolds. Instead of representing E ⊂ Rn by working in Rn, we work in the Grassmann Bundle, Rn × G(n, m). Advantages include, for example, the automatic encoding of tangent information directly into the representation. By building into the representation this tangent information, we make set comparisons where we care about tangent structure easy and natural.

Deﬁnition 5.11.1. The m-dimensional Grassmannian in Rn,

n G(n, m) = G(R , m), is the set of all m-dimensional planes through the origin.

For example, G(2, 1) is the space of all lines through the origin in R2, and G(3, 2) is the space of all planes through the origin in R3. The Grassmann bundle Rn × G(n, m) can be thought of as a space where G(n, m) is attached to each point in Rn.

Deﬁnition 5.11.2. A varifold is a Radon measure µ on the Grassmann bundle Rn × G(n, m).

Suppose π :(x, g) ∈ Rn × G(n, m) → x. One of the most common appearances of

n varifolds are those that arise from rectiﬁable sets E. In this case the measure µE on R × G(n, m) is the pushforward of m-Hausdorﬀ measure on E by the tangent map T : x →

(x, TxE).

Let E ⊂ Rn be an (Hm, m)-rectiﬁable set (see Deﬁnition 5.7.2). We know the approx-

m m imate m-dimensional tangent space TxE exists H -almost everywhere since E is (H , m)- rectiﬁable, which in turn implies that, except for an Hm-measure 0 set, E is contained in the union of the images of countably many Lipschitz functions from Rm to Rn. The measure of A ⊂ Rn × G(n, m) is given by µ(A) = Hm(T −1{A}). Let S ≡

{(x, TxE) | x ∈ E}, the section of the Grassmann bundle deﬁning the varifold. S, inter-

65 sected with each ﬁber {x} × G(n, m), is the single point (x, TxE), and so we could just as

m well use the projection π in which case we would have µE(A) = H (π(A ∩ S)).

m Definition 5.11.3. A rectifiable varifold is a radon measure µE defined on an (H , m)-

n n rectiﬁable set E ⊂ R . Recalling S ≡ {(x, TxE) | x ∈ E}, let A ⊂ R × G(n, m) and deﬁne

m µE(A) = H (π(A ∩ S)).

We will call E = π(S) the “downstairs” representation of S for any S ⊂ Rn × G(n, m), and we will call S = T (E) ⊂ Rn × G(n, m) the “upstairs” representation of any rectiﬁable set E, where T is the tangent map over the rectiﬁable set E.

π π S1 S2

π S π 2 2

2 2 0 R 0 R E2 E1 E

Figure 5.13: Working upstairs.

Figure 5.13, repeated from above, illustrates how the tangents are built into this rep-

resentation of subsets of Rn, giving us a sense of why this representation might be useful. Suppose we have three line segments almost touching each other, i.e. appearing to touch as

subsets of R2. The upstairs view puts each segment at a different height corresponding to the angle of the segment. So, these segments are not close in any sense in R2 × G(n, m). Or consider a straight line segment and a very fine sawtooth curve that may look practically indistinguishable, but will appear drastically different upstairs.

66 We can use varifold representations in combination with a cubical cover to get a quan- tized version of a curve that has tangent information as well as position information. If, for

2 1 example, we cover a set S ⊂ R × G(2, 1) with cubes of edge length 2d and use this cover as √ 3 a representation for S, we know the position and angle to within 2d+1 . In other words, we can approximate our curve S ⊂ R2 × G(2, 1) by the union of the centers of the cubes (with 1 edge length 2d ) intersecting S. This simple idea seems to merit further exploration.

5.12 Problems and Questions

Problem 5.12.1. Find a smooth ∂E, with E ⊂ Rn, such that

n−1 E n−1 H (∂Cd )/H (∂E) = 0 ∀d.

Hint: Look at unbounded E ⊂ R2 such that L2(Ec) < ∞.

Problem 5.12.2. Suppose that E is open and Hn−1(∂E) < ∞. Show that if the reach of ∂E is positive, then Hn−1(∂CE) lim inf d ≥ 1. d→∞ Hn−1(∂E)

Hint: First show that ∂E has unique inward and outward pointing normals. (Takes

a bit of work!) Next, examine the map F : ∂E → Rn, where F (x) = x + η(x)N(x), N(x) is the normal to ∂E at x, and η(x) is a positive real-valued function chosen so that locally

E F (∂E) = ∂Cd . Use the Binet-Cauchy Formula to ﬁnd the Jacobian, and then apply the Area

Formula. To do this calculation, notice that at any point x0 ∈ ∂E we can choose coordinates

n−1 n so that Tx0 ∂E is horizontal (i.e. N(x0) = en). Calculate using F : Tx0 ∂E = R → R where F (x) = x + η(x)N(x). (See Chapter 3 of [20] for the Binet-Cauchy formula and the Area Formula.)

Problem 5.12.3. Suppose E has dimension n − 1, positive reach, and is locally regular (in

Rn).

67 n E 1 a.) Find bounds for H (Cd )/ 2d . b.) How does this ratio relate to Hn−1(E)?

Hint: Use the ideas in Section 5.8 to calculate a bound on the volume of the tube with √ n thickness 2 2d centered on E.

Question 5.12.1. Can we use the “upstairs” version of cubical covers to ﬁnd better representations for sets and their boundaries? (Of course, “better” depends on your objective!)

For the following question, we need the notion of the multiscale ﬂat norm [48]. The basic idea of this distance, which works in spaces of oriented curves and surfaces of any dimension (known as currents), is that we can decompose the curve or surface T into (T −∂S)+∂S, but we measure the cost of the decomposition by adding the volumes of T − ∂S and S (not ∂S!). By volume, we mean the m-dimensional volume, or m-volume of an m-dimensional object, so if T is m-dimensional, we would add the m-volume of T − ∂S and the (m+1)-volume of S (scaled by the parameter λ). We get that

λ(T ) = min Mm(T − ∂S) + λMm+1(S). F S

It turns out that T − ∂S is the best approximation to T that has curvature bounded by λ [2]. We exploit this in the following ideas and questions.

Remark 5.12.1. Currents can be thought of as generalized oriented curves or surfaces of any dimension k. More precisely, they are members of the dual space to the space of k-forms. For the purposes of this section, thinking of them as (perhaps unions of pieces of) oriented k-dimensional surfaces W , so that W and −W are simply oppositely oriented and cancel if we add them, will be enough to understand what is going on. For a nice introduction to the ideas, see for example the ﬁrst few chapters of [47].

Question 5.12.2. Choose k ∈ {1, 2, 3}. In what follows we focus on sets Γ which are one- dimensional, the interior of a cube C will be denoted Co, and we will work at some scale d,

68 1 i.e. the edge length of the cube will be 2d . o o 1 Consider the piece of Γ in C , Γ ∩ C . Inside the cube C with edge length 2d , we will use the ﬂat norm to

1. ﬁnd an approximation of Γ ∩ Co with curvature bounded by λ = 2d+k and

2. ﬁnd the distance of that approximation from Γ ∩ Co.

This decomposition is then obtained by minimizing

o d+k 1 o d+k 2 M1((Γ ∩ C ) − ∂S) + 2 M2(S) = H ((Γ ∩ C ) − ∂S) + 2 L (S).

The minimal S will be denoted Sd (see Figure 5.14).

o o o Γ ∩ C ((Γ ∩ C ) − ∂Sd) ∩ C

F Figure 5.14: Multiscale ﬂat norm decomposition inspiring the deﬁnition of βΓ.

F Suppose that we deﬁne βΓ(C) by

F d+k 2 βΓ(C)l(C) = 2 L (Sd) so that

F 2d+k 2 βΓ(C) = 2 L (Sd).

69 What can we say about the properties (e.g. rectiﬁability) of Γ given the ﬁniteness of

2 X F βΓ(3C) l(C)? C

Question 5.12.3. Can we get an advantage by using the flat norm decomposition as a preconditioner before we find cubical cover approximations? For example, define

Γ Γd Fd ≡ Cd and Γd ≡ Γ − ∂Sd,

1 d+k 2 where Sd = argmin H (Γ − ∂S) + 2 L (S) . S

Since the ﬂat norm minimizers have bounded mean curvature, is this enough to force the cubical covers to give us better quantitative information on Γ? How about in the case in

which Γ = ∂E, E ⊂ R2?

70 CHAPTER 6

SUMMARY

The work herein develops a means for measuring the boundary of super-level sets in Rn under suitable conditions and explores the use of cubical covers to estimate the measure of certain “nice” sets in Rn. The main results involve the former and focus on the case where the boundary is C1,1, which implies positive reach (Definition 5.8.1), and the case in which the boundary is merely C1 and thus not necessarily of positive reach. The difference in the two cases is significant and requires separate approaches primarily because in the C1 case, we can have normals that intersect no matter how small of a distance we move outward along them. We begin by showing an exploration of a simple analysis problem (1.2.1) and the many directions one can work in to generalize and probe the result, leading to a deeper under- standing of what at first appears to be an elementary argument. We find that the same exponential functions serve as “fences” for our solution function to obtain the result in higher dimensions, Banach spaces, and even for functions that are Lipschitz (i.e. differentiable almost everywhere by Rademacher’s Theorem [20]; see Definition 1.4.1). We also note several observations that make the ratio ||Df||/||f|| peculiar, most importantly the fact that it detects roots by going off to infinity as we approach them. This fact then becomes the central characteristic used in the development of Theorem 3.1.1 and Theorem 4.1.1. In Theorem 3.1.1, we prove that the singular integral 3.1 computes the Hn−1-measure of the 0-level set of the function f ∈ C1,1, which is a C1,1 set by Theorem 2.2.1. Using the positive reach of this set, we can find an -neighborhood such that there exists a diffeomorphism between this neighborhood and the cylinder form by “lifting” the level set up or down by . Thus, we use the Area Formula (Theorem 2.2.3) along with the facts that Df is bounded on the closed ball B(0, 2R) and the Jacobian goes to 1 as → 0 to employ Fubini’s Theorem and obtain the desired result.

71 However, in Theorem 4.1.1, we no longer necessarily have a set of positive reach since our set A is only C1, so we use the distance function f to A, which has ||Df|| ≡ 1 almost everywhere, to show that our singular integral in (3.1) also computes the H1-measure of

A ⊂ R2, under suitable conditions. In this case, we use a “barehanded” approach by covering the (compact) set in rectangles, with comparatively very narrow cones containing the set A inside of each rectangle (by the geometric deﬁnition of the derivative [55], which exists for each a ∈ A since we have that A is a C1 set). By bounding the “overlap” of these rectangles so that this “overlap” goes to 0 before the width of the rectangles goes to 0, and by using the fact that the normals to A converge to the normals to center line of each rectangle, we can again use Fubini’s Theorem to obtain the desired result.

We next explore cubical covers of sets in Rn in a playful exposition designed as an invitation to further exploration of these ideas. We examine several basic results for bounding the measure of the cubical covers of the unions of balls, (n − 1)-rectiﬁable sets, and sets with a smooth (at least C1,1) boundary and positive reach. We also oﬀer a conjecture for a bound on the Hn−1-measure of the boundary of the cubical cover for a compact set with C1,1 boundary and provide a sketch of the proof for the n = 2 case.

The exposition continues with an introduction to another representation of sets in Rn, Jones’ β numbers, as a means of determining if a set E ∈ Rn lies in a rectifiable curve Γ (Theorem 5.10.1), which necessarily serves as a measure of how “flat” E looks as we zoom in on it since Γ has a linear approximation at H1-almost every point x ∈ Γ. Similarly, we introduce varifolds from geometric measure theory (Definition 5.11.2) which naturally incorporate information about the tangent space into the representation by attaching the m-dimensional Grassmannian to each point in Rn, which is simply the set of m-dimensional tangent planes through the origin in Rn. We call this space the Grassmann bundle and denote it by Rn × G(n, m). In this case, varifolds are then defined to be a radon measure µ on Rn × G(n, m) and, loosely speaking, allow us to examine our set “upstairs.” Lastly, we offer several problems and exercises for further exploration, especially via the multiscale flat

72 norm (5.12) from geometric measure theory, and also an extensive list of resources in the Appendix for Jones’ β numbers, current research in varifolds, and geometric measure theory, in general.

73 APPENDIX A.1 Further Exploration

There are a number of places to begin in exploring these ideas further. Some of these works require significant dedication to master, and it is always recommended that you have someone who has mastered a path into pieces of these areas that you can ask questions of when you first wade in. Nonetheless, if you remember that the language can always be translated into pictures, and you make the effort to do that, headway towards mastery can always be made. Here is an annotated list with comments:

Primary Varifold References Almgren’s little book [4] and Allard’s founding contribu- tion [1] are the primary sources for varifolds. Leon Simon’s book on geometric measure theory [52] (available for free online) has a couple of excellent chapters, one of which is an exposition of Allard’s paper.

Recent Varifold Work Both Buet and collaborators [9, 10,8, 11, 12] and Charon and collaborators [13, 14, 15] have been digging into varifolds with an eye to applications. While these papers are a good start, there is still a great deal of opportunity for the use and further development of varifolds. On the theoretical front, there is the work of Menne and collaborators [38, 39, 40, 41, 42, 30, 43, 44, 45, 46]. We want to call special attention to the recent introduction to the idea of a varifold that appeared in the December 2017 AMS Notices [45].

Geometric Measure Theory I The area underlying the ideas here are those from geometric measure theory. The fundamental treatise in the subject is still Federer’s 1969 Geometric Measure Theory [23] even though most people start by reading Mor- gan’s beautiful introduction to the subject, Geometric Measure Theory: A Beginner’s Guide [47] and Evans’ Measure Theory and Fine Properties of Functions [20]. Also recommended are Leon Simon’s lecture notes [52], Francesco Maggi’s book that up- dates the classic Italian approach [36], and Krantz and Parks’ Geometric Integration

75 Theory [31].

Geometric Measure Theory II The book by Mattila [37] approaches the subject from the harmonic-analysis-flavored thread of geometric measure theory. Some use this as a first course in geometric measure theory, albeit one that does not touch on minimal surfaces, which is the focus of the other texts above. De Lellis’ exposition Rectifiable Sets, Densities, and Tangent Measures [18] or Priess’ 1987 paper Geometry of Measures

in Rn: Distribution, Rectiﬁability, and Densities [50] is also very highly recommended.

Jones’ β In addition to the papers cited in the text [29, 49, 33, 51], there are related works by David and Semmes that we recommended. See for example [17]. There is also the applied work by Gilad Lerman and his collaborators that is often inspired by Jones’ β and his own development of Jones’ ideas in [33]. See also [16, 60, 59,6]. See also the work by Maggioni and collaborators [34,3].

Multiscale Flat Norm The ﬂat norm was introduced by Whitney in 1957 [58] and used to create a topology on currents that permitted Federer and Fleming, in their land- mark paper in 1960 [24], to obtain the existence of minimizers. In 2007, Morgan and Vixie realized that a variational functional introduced in image analysis was actually computing a multiscale generalization of the ﬂat norm [48]. The ideas are beginning to be explored in these papers [57, 19, 27, 56, 26,5].

A.2 Measures: A Brief Reminder

In this section we remind the reader of a handful of concepts used in the text.

Measure One way to think of a measure is as a generalization of the familiar notions of length, area, and volume in a way that allows us to deﬁne how we assign “size” to a given subset of a set X, the most common being that of n-dimensional Lebesgue measure Ln.

76 Formally, let X be a nonempty set and 2X be the collection of all subsets of X.A measure is deﬁned [20] to be a mapping µ : 2X → [0, ∞] such that

1. µ(∅) = 0 and

2. if

∞ A ⊂ ∪i=1Ai,

then ∞ X µ(A) ≤ µ(Ai). i=1 Note that in most texts, this deﬁnition is known as an outer measure, but we use this deﬁnition with the advantage that we can still “measure” non-measurable sets.

µ-measurable A subset S ⊂ X is called µ-measurable if and only if it satisﬁes the Carath´eodory condition for each set A ⊂ X:

µ(A) = µ(A ∩ S) + µ(A \ S).

Radon Measure Let us deﬁne the Borel sets in Rn to be those sets that are derived from the set of all open sets in Rn through the operations of countable union, countable intersection, and set diﬀerence. Then a measure µ on Rn is a Radon measure if [20]

1. every Borel set is µ-measurable; i.e. µ is a Borel measure.

2. for each A ⊂ Rn there exists a Borel set B such that A ⊂ B and µ(A) = µ(B); i.e. µ is Borel regular.

3. for each compact set K ∈ Rn, µ(K) < ∞; i.e. µ is locally ﬁnite.

77 Approximate Tangent Plane We present here an approximate tangent k-plane based on integration. (The one-dimensional version is of course an approximate tangent line.)

We start with the fact that we can integrate functions deﬁned on Rn over k-dimensional sets using k-dimensional measures µ (typically Hk). We zoom in on the point p through dilation of the set F :

y − p F (p) = {x ∈ n | x = + p for some y ∈ F }. ρ R ρ

We will say that the set F has an approximate tangent k-plane L at p if the dilation

of Fρ(p) converges weakly to L; i.e. if

Z Z φdµ → φdµ as ρ → 0 Fρ L

for all continuously diﬀerentiable, compactly supported φ : Rn → R.

In the next two ﬁgures, we note that the solid green lines are the level sets of φ while the dashed green line indicates the boundary of the support of φ. Note also that the ρ’s of 0.4, 0.1, and 0.02 are approximate.

L F φ p 0.02

F0.1 F0.4

Figure A.1: The case of 1-planes (lines) where L is the weak limit of the dilations of F . (Image used with permission from Kevin R. Vixie.)

78 L

F φ p 0.02

F0.1 F0.4

Figure A.2: The case of 1-planes (lines) where L is not the weak limit of the dilations of F . (Image used with permission from Kevin R. Vixie.)

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