Strange phenomena in convex and discrete geometry
著者
書誌事項
Strange phenomena in convex and discrete geometry
（Universitext）
Springer, c1996
- : soft
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注記
Includes bibliographical references (p. [143]-153) and index
内容説明・目次
内容説明
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.
目次
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 Toth-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.
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