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AuthorZong, Chuanming. author
TitleStrange Phenomena in Convex and Discrete Geometry [electronic resource] / by Chuanming Zong ; edited by James J. Dudziak
ImprintNew York, NY : Springer New York, 1996
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Descript VI, 158p. 10 illus. online resource


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


1 Borsukโ{128}{153}s Problem -- ยง1 Introduction -- ยง2 The Perkal-Eggleston Theorem -- ยง3 Some Remarks -- ยง4 Larmanโ{128}{153}s Problem -- ยง5 The Kahn-Kalai Phenomenon -- 2 Finite Packing Problems -- ยง1 Introduction -- ยง2 Supporting Functions, Area Functions, Minkowski Sums, Mixed Volumes, and Quermassintegrals -- ยง3 The Optimal Finite Packings Regarding Quermassintegrals -- ยง4 The L. Fejes Tรณth-Betke-Henk-Wills Phenomenon -- ยง5 Some Historical Remarks -- 3 The Venkov-McMullen Theorem and Steinโ{128}{153}s Phenomenon -- ยง1 Introduction -- ยง2 Convex Bodies and Their Area Functions -- ยง3 The Venkov-McMullen Theorem -- ยง4 Steinโ{128}{153}s Phenomenon -- ยง5 Some Remarks -- 4 Local Packing Phenomena -- ยง1 Introduction -- ยง2 A Phenomenon Concerning Blocking Numbers and Kissing Numbers -- ยง3 A Basic Approximation Result -- ยง4 Minkowskiโ{128}{153}s Criteria for Packing Lattices and the Densest Packing Lattices -- ยง5 A Phenomenon Concerning Kissing Numbers and Packing Densities -- ยง6 Remarks and Open Problems -- 5 Category Phenomena -- ยง1 Introduction -- ยง2 Gruberโ{128}{153}s Phenomenon -- ยง3 The Aleksandrov-Busemann-Feller Theorem -- ยง4 A Theorem of Zamfirescu -- ยง5 The Schneider-Zamfirescu Phenomenon -- ยง6 Some Remarks -- 6 The Busemann-Petty Problem -- ยง1 Introduction -- ยง2 Steiner Symmetrization -- ยง3 A Theorem of Busemann -- ยง4 The Larman-Rogers Phenomenon -- ยง5 Schneiderโ{128}{153}s Phenomenon -- ยง6 Some Historical Remarks -- 7 Dvoretzkyโ{128}{153}s Theorem -- ยง1 Introduction -- ยง2 Preliminaries -- ยง3 Technical Introduction -- ยง4 A Lemma of Dvoretzky and Rogers -- ยง5 An Estimate for ?V(AV) -- ยง6 ?-nets and ?-spheres -- ยง7 A Proof of Dvoretzkyโ{128}{153}s Theorem -- ยง8 An Upper Bound for M (n, ?) -- ยง9 Some Historical Remarks -- Inedx

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