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.
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 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.
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.
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