Strange Phenomena in Convex and Discrete Geometry

ISBN-13: 9780387947341 / Angielski / Miękka / 1996 / 158 str.

Chaunming Zong; J. J. Dudziak; C. Zong

Opis

Convex and discrete geometry is one of the most intuitive subjects in mathematics. One can explain many of its problems, even the most difficult - such as the sphere-packing problem (what is the densest possible arrangement of spheres in an n-dimensional space?) and the Borsuk problem (is it possible to partition any bounded set in an n-dimensional space into n+1 subsets, each of which is strictly smaller in "extent" than the full set?) - in terms that a layman can understand; and one can reasonably make conjectures about their solutions with little training in mathematics.

Spis treści

1 Borsuk’s Problem.- §1 Introduction.- §2 The Perkal-Eggleston Theorem.- §3 Some Remarks.- §4 Larman’s Problem.- §5 The Kahn-Kalai Phenomenon.- 2 Finite Packing Problems.- §1 Introduction.- §2 Supporting Functions, Area Functions, Minkowski Sums, Mixed Volumes, and Quermassintegrals.- §3 The Optimal Finite Packings Regarding Quermassintegrals.- §4 The L. Fejes Tóth-Betke-Henk-Wills Phenomenon.- §5 Some Historical Remarks.- 3 The Venkov-McMullen Theorem and Stein’s Phenomenon.- §1 Introduction.- §2 Convex Bodies and Their Area Functions.- §3 The Venkov-McMullen Theorem.- §4 Stein’s Phenomenon.- §5 Some Remarks.- 4 Local Packing Phenomena.- §1 Introduction.- §2 A Phenomenon Concerning Blocking Numbers and Kissing Numbers.- §3 A Basic Approximation Result.- §4 Minkowski’s Criteria for Packing Lattices and the Densest Packing Lattices.- §5 A Phenomenon Concerning Kissing Numbers and Packing Densities.- §6 Remarks and Open Problems.- 5 Category Phenomena.- §1 Introduction.- §2 Gruber’s Phenomenon.- §3 The Aleksandrov-Busemann-Feller Theorem.- §4 A Theorem of Zamfirescu.- §5 The Schneider-Zamfirescu Phenomenon.- §6 Some Remarks.- 6 The Busemann-Petty Problem.- §1 Introduction.- §2 Steiner Symmetrization.- §3 A Theorem of Busemann.- §4 The Larman-Rogers Phenomenon.- §5 Schneider’s Phenomenon.- §6 Some Historical Remarks.- 7 Dvoretzky’s Theorem.- §1 Introduction.- §2 Preliminaries.- §3 Technical Introduction.- §4 A Lemma of Dvoretzky and Rogers.- §5 An Estimate for ?V(AV).- §6 ?-nets and ?-spheres.- §7 A Proof of Dvoretzky’s Theorem.- §8 An Upper Bound for M (n, ?).- §9 Some Historical Remarks.- Inedx.