James J. Dudziak

Strange Phenomena in Convex and Discrete Geometry

1 Borsuks Problem.- x1 Introduction.- x2 The Perkal-Eggleston Theorem.- x3 Some Remarks.- x4 Larmans Problem.- x5 The Kahn-Kalai Phenomenon.- 2 Finite Packing Problems.- x1 Introduction.- x2 Supporting Functions, Area Functions, Minkowski Sums, Mixed Volumes, and Quermassintegrals.- x3 The Optimal Finite Packings Regarding Quermassintegrals.- x4 The L. Fejes Toth-Betke-Henk-Wills Phenomenon.- x5 Some Historical Remarks.- 3 The Venkov-McMullen Theorem and Steins Phenomenon.- x1 Introduction.- x2 Convex Bodies and Their Area Functions.- x3 The Venkov-McMullen Theorem.- x4 Steins Phenomenon.- x5 Some Remarks.- 4 Local Packing Phenomena.- x1 Introduction.- x2 A Phenomenon Concerning Blocking Numbers and Kissing Numbers.- x3 A Basic Approximation Result.- x4 Minkowskis Criteria for Packing Lattices and the Densest Packing Lattices.- x5 A Phenomenon Concerning Kissing Numbers and Packing Densities.- x6 Remarks and Open Problems.- 5 Category Phenomena.- x1 Introduction.- x2 Grubers Phenomenon.- x3 The Aleksandrov-Busemann-Feller Theorem.- x4 A Theorem of Zamfirescu.- x5 The Schneider-Zamfirescu Phenomenon.- x6 Some Remarks.- 6 The Busemann-Petty Problem.- x1 Introduction.- x2 Steiner Symmetrization.- x3 A Theorem of Busemann.- x4 The Larman-Rogers Phenomenon.- x5 Schneiders Phenomenon.- x6 Some Historical Remarks.- 7 Dvoretzkys Theorem.- x1 Introduction.- x2 Preliminaries.- x3 Technical Introduction.- x4 A Lemma of Dvoretzky and Rogers.- x5 An Estimate for ?V(AV).- x6 ?-nets and ?-spheres.- x7 A Proof of Dvoretzkys Theorem.- x8 An Upper Bound for M (n, ?).- x9 Some Historical Remarks.- Inedx.

Vitushkin's conjecture for removable sets

Preface.- 1 Removable Sets and Analytic Capacity.- 1.1 Removable Sets.- 1.2 Analytic Capacity.- 2 Removable Sets and Hausdor Measure.- 2.1 Hausdor Measure and Dimension.- 2.2 Painleves Theorem.- 2.3 Frostmans Lemma.- 2.4 Conjecture & Refutation: The Planar Cantor Quarter Set.- 3 Garabedian Duality for Hole-Punch Domains.- 3.1 Statement of the Result and an Initial Reduction.- 3.2 Interlude: Boundary Correspondence for H1(U).- 3.3 Interlude: Some F. & M. Riesz Theorems.- 3.4 Construction of the Boundary Garabedian Function.- 3.5 Construction of the Interior Garabedian Function.- 3.6 A Further Reduction.- 3.7 Interlude: Some Extension and Join Propositions.- 3.8 Analytically Extending the Ahlfors and Garabedian Functions.- 3.9 Interlude: Consequences of the Argument Principle.- 3.10 An Analytic Logarithm of the Garabedian Function.- 4 Melnikov and Verderas Solution to the Denjoy Conjecture.- 4.1 Menger Curvature of Point Triples.- 4.2 Melnikovs Lower Capacity Estimate.- 4.3 Interlude: A Fourier Transform Review.- 4.4 Melnikov Curvature of Some Measures on Lipschitz Graphs.- 4.5 Arclength & Arclength Measure: Enough to Do the Job.- 4.6 The Denjoy Conjecture Resolved Affirmatively.- 4.7 Conjecture & Refutation: The Joyce-Morters Set.- 5 Some Measure Theory.- 5.1 The Caratheodory Criterion and Metric Outer Measures.- 5.2 Arclength & Arclength Measure: The Rest of the Story.- 5.3 A Vitali Covering Lemma and Planar Lebesgue Measure.- 5.4 Regularity Properties of Hausdor Measures.- 5.5 The Besicovitch Covering Lemma and Lebesgue Points.- 6 A Solution to Vitushkins Conjecture Modulo Two Difficult Results.- 6.1 Statement of the Conjecture and a Reduction.- 6.2 Cauchy Integral Representation.- 6.3 Estimates of Truncated Cauchy Integrals.- 6.4 Estimates of Truncated Suppressed Cauchy Integrals.- 6.5 Vitushkins Conjecture Resolved Affirmatively Modulo Two Difficult Results.- 6.6 Postlude: The Original Vitushkin Conjecture.- 7 The T(b) Theorem of Nazarov, Treil, and Volberg.- 7.1 Restatement of the Result.- 7.2 Random Dyadic Lattice Construction.- 7.3 Lip(1)-Functions Attached to Random Dyadic Lattices.- 7.4 Construction of the Lip(1)-Function of the Theorem.- 7.5 The Standard Martingale Decomposition.- 7.6 Interlude: The Dyadic Carleson Imbedding Inequality.- 7.7 The Adapted Martingale Decomposition.- 7.8 Bad Squares and Their Rarity.- 7.9 The Good/Bad-Function Decomposition.- 7.10 Reduction to the Good Function Estimate.- 7.11 A Sticky Point, More Reductions, and Course Setting.- 7.12 Interlude: The Schur Test.- 7.13 G1: The Crudely Handled Terms.- 7.14 G2: The Distantly Interacting Terms.- 7.15 Splitting Up the G3 Terms.- 7.16 Gterm 3 : The Suppressed Kernel Terms.- 7.17 Gtran 3 : The Telescoping Terms.- 8 The Curvature Theorem of David and Leger.- 8.1 Restatement of the Result and an Initial Reduction.- 8.2 Two Lemmas Concerning High Density Balls.- 8.3 The Beta Numbers of Peter Jones.- 8.4 Domination of Beta Numbers by Local Curvature.- 8.5 Domination of Local Curvature by Global Curvature.- 8.6 Selection of Parameters for the Construction.- 8.7 Construction of a Baseline L0.- 8.8 De nition of a Stopping-Time Region S0.- 8.9 De nition of a Lipschitz Set K0 over the Base Line.- 8.10 Construction of Adapted Dyadic Intervals on the Base Line.- 8.11 Assigning Linear Functions to Adapted Dyadic Intervals.- 8.12 Construction of a Lipschitz Graph G Threaded through K0.- 8.13 Veri cation that the Graph is Indeed Lipschitz.- 8.14 A Partition of K n K0 into Three Sets: K1, K2, & K3.- 8.15 The Smallness of the Set K2.- 8.16 The Smallness of a Horrible Set H.- 8.17 Most of K Lies in the Vicinity of the Lipschitz Graph.- 8.18 The Smallness of the Set K1.- 8.19 Gamma Functions of the Lipschitz Graph.- 8.20 A Point Estimate on One of the Gamma Functions.- 8.21 A Global Estimate on the Other Gamma Function.- 8.22 Interlude:

Hankel Operators on Bounded Analytic Functions

For U a domain in the complex plane and g a bounded measurable function on U , the generalized Hankel operator Sg on H∞(U) is the operator of multiplication by g followed by projection into L∞/H∞. Under certain conditions on U we show that either Sg is compact or there is an embedded `∞ on which Sg is bicontinuous. We characterize those g’s for which Sg is compact in the case that U is a Behrens roadrunner domain.

Von Neumann operators are reflexive

A compact subset K of the complex plane is a spectral set for an operator T on a Hubert space Jif if the spectrum of T, σ (T), is contained in K and for every rational function / with poles off K, \ \ f ( T ) \ \ ^ \ \ f \ \ K = sup{\f(z)\:zeK}. The operator T is called a von Neumann operator if the spectrum of T is itself a spectral set for T. It has been known for some time that a von Neumann operator has a non-trivial invariant subspace [1]. In this paper it will be shown that every von Neumann operator is reflexive. That is, if A is any operator such that AJi^M for every subspace M of 3f that is invariant for Γ, then A belongs to the weakly closed operator algebra generated by T and the identity.

The Busemann-Petty Problem

The search for relationships between a convex body and its projections or sections has a long history. In 1841, A. Cauchy found that the surface area of a convex body can be expressed in terms of the areas of its projections as follows:

The Curvature Theorem of David and Léger

s\left( K \right) = \frac{1}{{{\omega _{n - 1}}}}\int_{\partial \left( B \right)} {\bar v\left( {{P_u}\left( K \right)} \right)d\lambda \left( u \right)} .

Removable Sets and Hausdorff Measure

Here, s(K) denotes the surface area of a convex body K ⊂ R n , \(\bar v\left( X \right)\) denotes the (n − 1)-dimensional “area” of a set X ⊂ R n −1, P u denotes the orthogonal projection from R n to the hyperplane H u = {x ∈ R n : 〈x, u〉 = 0} determined by a unit vector u of R n , and λ denotes surface-area measure on ∂(B). In contrast, the closely related problem of finding an expression for the volume of K in terms of the areas of its projections P u (K) (or the areas of its sections I u (K) = K ⋂ H u ) proved to be unexpectedly and extremely difficult.

Borsuk’s Problem

Let ,u be a positive Borel measure on a compact subset K of the complex plane. Denote the weak-star closure in L? (i) of R(K) by ROO(K,u). Given f E RI(K,u), denote the weak-star closure in LI(p) of the algebra generated by R??(K,j) and the complex conjugate of f by AI (f, i) . This paper determines the structure of AI (f, ) . As a consequence, a solution is obtained to a problem concerned with minimal normal extensions of functions of a subnormal operator.