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Polar - Legendre Duality In POLAR - LEGENDRE DUALITY IN CONVEX GEOMETRY AND GEOMETRIC FLOWS A Thesis Presented To The Academic Faculty By Edward C. White, Jr. In Partial Fulfillment Of the Requirements for the Degree Master of Science in Mathematics Georgia Institute of Technology August, 2008 Copyright c Edward C. White, Jr., 2008 POLAR - LEGENDRE DUALITY IN CONVEX GEOMETRY AND GEOMETRIC FLOWS Approved by: Dr. Evans Harrell, Advisor School of Mathematics Georgia Institute of Technology Dr. Mohammad Ghomi School of Mathematics Georgia Institute of Technology Dr. Guillermo Goldsztein School of Mathematics Georgia Institute of Technology Date Approved: June 26, 2008 ACKNOWLEDGEMENTS I wish first to acknowledge the help of my thesis advisor, Dr. Evans Harrell at Georgia Institute of Technology, whose insights, encouragement and deep appreciation for analysis and geometry were critical factors behind this thesis. Thanks are also due to Dr. Mohammad Ghomi and Dr. Guillermo Goldsztein, both at Georgia Institute of Technology, who sat on my thesis committee and provided many helpful comments and suggestions. I also owe a debt of gratitude to my wife, Irma M. White, and all of our family and friends for their patience as I divided my time, not always in a balanced way, between them and my scholarly pursuits. iii TABLE OF CONTENTS ACKNOWLEDGEMENTS.................................................................................iii LIST OF TABLES..............................................................................................vii LIST OF FIGURES...........................................................................................viii LIST OF SYMBOLS...........................................................................................ix SUMMARY.........................................................................................................x Chapter 1: INTRODUCTION..............................................................................1 Chapter 2: POLAR DUAL, SUPPORT AND GAUGE FUNCTION....................3 2.1 The polar dual..............................................................................3 2.2 The support function....................................................................5 2.2.1 The intuitive support function.............................................6 2.2.2 Representation of a convex body........................................11 2.3 The gauge function.....................................................................12 2.3.1 Characterizing the gauge function.....................................12 2.3.2 Representation of a convex body........................................13 2.4 Polar and Legendre duality.........................................................15 2.4.1 A theorem on polar duality................................................19 2.4.2 A theorem on Legendre duality..........................................22 2.5 Case study: the ellipse................................................................26 iv TABLE OF CONTENTS (CONT.) 2.5.1 The polar dual.....................................................................27 2.5.2 Support function.................................................................28 2.5.3 Gauge function....................................................................30 2.5.4 Support function of the dual................................................31 2.5.5 Gauge function of the dual...................................................32 2.5.6 Verification of polar duality..................................................33 2.5.7 Generalized Legendre Transform..........................................34 Chapter 3 SUPPORT FUNCTIONS AND GEOMETRIC FLOWS.....................40 3.1 General results.............................................................................40 3.2 Harmonic mean curvature flow.....................................................50 3.3 Inverse harmonic mean curvature flow..........................................51 3.4 Mean curvature flow.....................................................................52 3.5 Gauss curvature flow....................................................................52 3.6 Affine normal flow........................................................................53 3.7 Ricci curvature flow......................................................................53 Chapter 4 GEOMETRIC FLOWS AND POLAR DUALITY..............................56 4.1 Background..................................................................................56 4.2 The case when n = 2.....................................................................59 4.3 The case when n = 3....................................................................70 v TABLE OF CONTENTS (CONT.) Chapter 5 MAHLER'S CONJECTURE..............................................................78 5.1 The conjecture.............................................................................78 5.2 The intuitive geometric interpretation..........................................80 5.3 Partial results known to date........................................................83 5.3.1 Result 1: Zonoids................................................................84 5.3.2 Result 2: True up to a factor of 2−n......................................89 5.3.3 A table of known results.......................................................91 Chapter 6 DUALITY AND MAHLER'S CONJECTURE...................................93 6.1 Extending Beyond Two Dimensions.............................................96 Chapter 7 CONCLUSION...................................................................................97 REFERENCES...................................................................................................101 vi LIST OF TABLES Table 1 A Subtlety in Polar Duality..................................................................22 Table 2 Geometric Flows and Their Dual Flows...............................................69 Table 3 Known Mahler's Conjecture Results....................................................92 vii LIST OF FIGURES Figure 1. Support hyperplanes to a convex body A.............................................8 Figure 2. The support function...........................................................................9 Figure 3. The support function as a distance.....................................................10 Figure 4. A representation of the gauge function of a convex body......................14 Figure 5. Polar - Legendre Commutative Diagram.............................................18 Figure 6. Representation of an ellipse A and its polar dual A∗............................29 Figure 7. The immersion map and the support function....................................47 Figure 8. The relationship between φ and θ.......................................................60 hθ Figure 9. The geometry of the logarithmic derivative h ...................................63 Figure 10. Filled join, nested between hyperbola..............................................90 viii LIST OF SYMBOLS * Polar duality. A∗, for example, is the polar dual of A. g(x) The gauge function of a convex body. r In 2 or 3 dimensions, the gradient. r In higher dimensions, the Levi-Civita connection. r¯ The gradient on the unit n-dimensional ball. h(u) The support function of a convex body. hθ; hθθ The first and second derivatives of h, with respect to θ. K4 The filled join of a convex body K. 4¯ The Laplacian on the unit sphere. L The (Generalized) Legendre Transform. φ(x) An immersion describing a convex body. u A vector (any letter in boldface). ^u A unit normal vector to a the surface of a convex body. V (A) Volume of a convex body A. VP (A) Volume Product: V (A) ∗ V (A∗). ix SUMMARY This thesis examines the elegant theory of polar and Legendre duality, and its potential use in convex geometry and geometric analysis. As a starting point, it derives a theorem of polar - Legendre duality for all convex bodies, independent of the degree of smoothness or strict convexity of their bound- aries. This theorem is captured in a commutative diagram that shows how it is possible to transform information about a convex body into information about its polar dual, and vice versa. The notion of a geometric flow on a convex body inducing a flow on a polar dual is introduced, and equations for the original and the dual flow are worked out for a number of two dimensional flows. It is shown that in general these flows are not geometric, in the sense of being defined solely by local curvature, but the evolution equations, at least in two dimensions, have similarities to the inverse flows on the original convex bodies. This theory is extended, in a preliminary way, to three dimensions. Mahler's Conjecture in convex geometry is discussed in some detail, and polar-Legendre duality is used to re-examine the resolved two-dimensional problem. This leads to some ideas on ways to attack the still open three-dimensional conjecture. x As part of this work the ratio of the derivative of a support function to the hθ support function, better known as the logarithmic derivative h , is examined analytically and geometrically. It is a ratio that is important in both two- dimensional and three-dimensional geometric flows. The logarithmic second hθθ derivative, h , is important in Mahler's Conjecture. xi CHAPTER 1 Introduction Duality is one of the most powerful and elegant concepts in modern math- ematics. It is powerful because it can be used to present two ways of ap- proaching difficult problems, often with dramatic results. It is elegant
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