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.

Inhalt

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