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BOUNDARY MEASURES and CUBICAL COVERS of SETS in Rn

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BOUNDARY MEASURES and CUBICAL COVERS of SETS in Rn BOUNDARY MEASURES AND CUBICAL COVERS OF SETS IN Rn By 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 c Copyright by LARAMIE SMITH PAXTON, 2018 All Rights Reserved c Copyright by LARAMIE SMITH PAXTON, 2018 All Rights Reserved To the Faculty of Washington State University: 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 math- ematics. 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 jjDfjj=jjfjj, 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 2 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) < 1 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)
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