Chuanming Zong

Strange Phenomena in Convex and Discrete Geometry (eBook, PDF)

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Springer New York
• Seitenzahl: 158
• Erscheinungstermin: 6. Dezember 2012
• Englisch
• ISBN-13: 9781461384816
• Artikelnr.: 43987485

Inhaltsangabe

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.