Strange Phenomena in Convex and Discrete Geometry

1 Borsuk's Problem.- x1 Introduction.- x2 The Perkal-Eggleston Theorem.- x3 Some Remarks.- x4 Larman's 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 Stein's Phenomenon.- x1 Introduction.- x2 Convex Bodies and Their Area Functions.- x3 The Venkov-McMullen Theorem.- x4 Stein's 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 Minkowski's 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 Gruber's 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 Schneider's Phenomenon.- x6 Some Historical Remarks.- 7 Dvoretzky's 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 Dvoretzky's Theorem.- x8 An Upper Bound for M (n, ?).- x9 Some Historical Remarks.- Inedx.