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DUALITY PHENOMENA and VOLUME INEQUALITIES in CONVEX GEOMETRY a Dissertation Submitted to Kent State University in Partial Fulfil DUALITY PHENOMENA AND VOLUME INEQUALITIES IN CONVEX GEOMETRY A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Jaegil Kim August, 2013 Dissertation written by Jaegil Kim B.S., Pohang University of Science and Technology, 2005 M.S., Pohang University of Science and Technology, 2008 Ph.D., Kent State University, 2013 Approved by Chair, Doctoral Dissertation Committee Dr. Artem Zvavitch Members, Doctoral Dissertation Committee Dr. Richard M. Aron Members, Doctoral Dissertation Committee Dr. Fedor Nazarov Members, Doctoral Dissertation Committee Dr. Dmitry Ryabogin Members, Outside Discipline Dr. Fedor F. Dragan Members, Graduate Faculty Representative Dr. David W. Allender Accepted by Chair, Department of Mathematical Sciences Dr. Andrew Tonge Associate Dean Raymond Craig Dean, College of Arts and Sciences ii TABLE OF CONTENTS ACKNOWLEDGEMENTS . v Introduction and Preliminaries 1 0.1 Basic concepts in Convex Geometry . 2 0.2 Summary and Notation . 5 0.3 Bibliographic Note . 7 I Mahler conjecture 8 1 Introduction to the volume product . 9 1.1 The volume product and its invariant properties . 9 1.2 Mahler conjecture and related results . 13 2 More on Hanner polytopes . 16 2.1 Connection with perfect graphs . 16 2.2 Direct sums of polytopes . 20 3 Local minimality in the symmetric case . 31 3.1 Construction of Polytopes . 31 3.2 Computation of volumes of polytopes . 41 3.3 A gap between K and a polytope . 60 4 Local minimality in the nonsymmetric case . 79 4.1 Continuity of the Santal´omap . 80 iii 4.2 Construction of polytopes for the simplex case . 86 5 Stability results for unconditional bodies . 94 5.1 Stability for unconditional bodies . 95 5.2 Minimality near unconditional convex bodies . 103 6 Proposed further research . 106 II `Convexity' of intersection bodies 108 7 Introduction to intersection bodies . 109 7.1 Classical theorems and questions . 110 7.2 Several ways to measure `convexity' . 111 8 Quasi-convexity for intersection bodies . 114 8.1 Quasi-convexity and related results . 114 8.2 Generalization to log-concave measures . 122 8.3 Non-symmetric cases and s-concave measures . 127 9 Local convexity for bodies of revolution . 132 9.1 Equatorial power type for bodies of revolution . 133 9.2 Equatorial power type 2 for intersection bodies . 136 9.3 Double intersection bodies of revolution in high dimension . 149 10 Uniform convexity for intersection bodies . 155 10.1 The Proof of Theorem . 157 11 Proposed further research . 166 BIBLIOGRAPHY . 167 iv ACKNOWLEDGEMENTS First of all I would like to thank my advisor Dr. Artem Zvavitch for his constant support and guidance throughout my graduate study. He has been a great mentor not only in my academic pursuits but also in my life at Kent State. Next I would like to thank Dr. Richard Aron, Dr. Fedor Nazarov, and Dr. Dmitry Ryabogin for their help in class or seminar and also serving as my dissertation committee members. Lastly I would like to thank my family, my wife Sieun and my son Sonu for their support and love. v Introduction and Preliminaries 1 2 We start with the review of some classical background and general techniques, containing John's Theorem and the Brunn-Minkowski Inequality, in Convex Geometry. We refer the reader to the books [22], [47], [48], [68], [74] for more information on classical results and their application in Convex Geometry and Geometric Tomography. 0.1 Basic concepts in Convex Geometry n We denote by R the Euclidean space of dimension n with the inner product ; and h· ·i n the standard basis e1; : : : ; en . A nonempty subset S of R is said to be f g convex if the midpoint of any two points in S belongs to S. • symmetric if it is symmetric at the origin, i.e. x S for all x S. • − 2 2 unconditional (with respect to a basis e1; : : : ; en ) if it is symmetric with respect to • f g n each coordinate hyperplane, i.e., tj x; ej S for any x S and any choice of 1 h i 2 2 signs t1; : : : ; tn 1; 1 . P 2 {− g star-shaped at a point p S if the intersection with any straight line containing p is • 2 always a line segment. n For a star-shaped set K at the origin, the Minkowski functional of K is defined in R by n x K = min λ > 0 : x λK ; x R k k f 2 g 2 and the radial function of K is defined on the sphere Sn−1 by ρ (u) = max λ > 0 : λu K = u −1; u Sn−1: K f 2 g k kK 2 n n A body in R is a compact set in R which is the closure of its interior. In particular, a convex body is a body which is convex, or, equivalently, a compact convex set with nonempty 3 interior. A star body is a star-shaped body at the origin whose radial function is positive and continuous. For example, the n-dimensional `p ball, which is defined by n n n p Bp = x R : x; ej 1 when 0 < p < 2 j=1 j h i j ≤ 1 n o n n X B1 = x R : x; ej 1 for 1 j n when p = ; 2 jh ij ≤ ≤ ≤ 1 n o is a star body if p > 0, and is a convex body if p 1. ≥ n Let K be a convex body in R containing the origin in its interior. The polar body of K is defined by ◦ n K = y R x; y 1 for all x K : 2 h i ≤ 2 This concept of duality (polarity) in convex bodies can be generalized by taking another n interior point instead of the origin. If K R is a convex body and z is an interior point ⊂ of K, then the polar body of K with respect to z, denoted by Kz, is given as z n K = y R : y z; x z 1 for all x K : f 2 h − − i ≤ 2 g It can be expressed, in terms of the usual polarity, as Kz = (K z)◦ + z. Moreover we have − the following basic properties of polarity. n n n Proposition 0.1. Let K, L be convex bodies in R . Let T : R R be an invertible ! linear operator, and fix u Sn−1. 2 1. (K◦)◦ = K (Kz)z = K 2. K◦ L◦ whenever K LKz Lz whenever K L ⊃ ⊂ ⊃ ⊂ 3. (K L)◦ = conv(K◦;L◦)(K L)z = conv(Kz;Lz) \ \ 4. (K u?)◦ = K◦ u? where K u? denotes the projection of K to u?. \ j j 5. (TK)◦ = (T ∗)−1K◦ where T ∗ denotes the adjoint operator of T . See [22], [47], [48], and [79] for the proof of the above properties. 4 One of fundamental results in Convex Geometry is the Brunn-Minkowski inequality. It will be useful for us to prove some of results related to the volume of a convex body. Theorem 0.1 (Brunn-Minkowski inequality). Let, A; B be non-empty compact subsets of n R . Then, A + B 1=n A 1=n + B 1=n; j j ≥ j j j j or, equivalently, for all λ [0; 1], 2 λA + (1 λ)B A λ B 1−λ; j − j ≥ j j j j n where A + B = a + b R : a A; b B is the Minkowski sum of A and B, and λA = f 2 2 2 g n λa R : a A . f 2 2 g In order to study local properties of convex bodies, we may consider two different metrics among convex bodies: the Banach-Mazur distance and the Hausdorff distance. The classical Banach-Mazur distance dBM between two symmetric bodies is defined by dBM (K; L) = min r 1 : L TK rL for T GL(n) ; ≥ ⊂ ⊂ 2 n o n where GL(n) is the set of all invertible linear transformations on R . More generally, in case that we do not have symmetry condition on bodies K and L, it can be defined as dBM (K; L) = min r 1 : AL BK rAL for A; B AL(n) ; ≥ ⊂ ⊂ 2 n o n where AL(n) is the set of all invertible affine transformations on R . In addition, the Hausdorff distance dH between two bodies K and L is defined by dH(K; L) = max max min x y ; max min x y ; x2K y2L j − j y2L x2K j − j 5 or, equivalently, n n dH(K; L) = min " 0 : K L + "B ;L K + "B : ≥ ⊂ 2 ⊂ 2 n o n We note that the volume function of bodies in R is continuous with respect to both the Banach-Mazur distance and the Hausdorff distance. Moreover the Banach-Mazur distance satisfies the following classical theorem by F. John. n Theorem 0.2 (John). For any convex symmetric body K R , we get ⊂ n dBM (K; B ) pn: 2 ≤ Here the equality holds if and only if K is an ellipsoid. One of the consequences of the theorem is that the set of symmetric convex bodies in n R is compact with respect to the Banach-Mazur distance. 0.2 Summary and Notation In this dissertation we investigate two topics related to the volume in Convex Geometry; one is about duality phenomena, and the other is about intersection bodies. First, the volume product of a symmetric convex body is defined as the product of volumes of the body and its polar body. The Blaschke-Santalo inequality says that the maximum of the volume product is attained at the Euclidean balls.
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