Handbook of Convex Geometry, Volume B offers a survey of convex geometry and its many ramifications and connections with other fields of mathematics, including convexity, lattices, crystallography, and convex functions.

The selection first offers information on the geometry of numbers, lattice points, and packing and covering with convex sets. Discussions focus on packing in non-Euclidean spaces, problems in the Euclidean plane, general convex bodies, computational complexity of lattice point problem, centrally symmetric convex bodies, reduction theory, and lattices and the space of lattices. The text then examines finite packing and covering and tilings, including plane tilings, monohedral tilings, bin packing, and sausage problems.

The manuscript takes a look at valuations and dissections, geometric crystallography, convexity and differential geometry, and convex functions. Topics include differentiability, inequalities, uniqueness theorems for convex hypersurfaces, mixed discriminants and mixed volumes, differential geometric characterization of convexity, reduction of quadratic forms, and finite groups of symmetry operations.

The selection is a dependable source of data for mathematicians and researchers interested in convex geometry.

Contenu

VOLUME B. Preface. Part 3: Discrete Aspects of Convexity. Geometry of Numbers (P.M. Gruber). Lattice points (P. Gritzmann, J.M. Wills). Packing and covering with convex sets (G. Fejes Tóth, W. Kuperberg). Finite packing and covering (P. Gritzmann, J.M. Wills). Tilings (E. Schulte). Valuations and dissections (P. McMullen). Geometric crystallography (P. Engel). Part 4: Analytic Aspects of Convexity. Convexity and differential geometry (K. Leichtweiss). Convex functions (A.W. Roberts). Convexity and calculus of variations (U. Brechtken-Manderscheid, E. Heil). On isoperimetric theorems of mathematical physics (G. Talenti). The local theory of normed spaces and its applications to convexity (J. Lindenstrauss, V. Milman). Nonexpansive maps and fixed points (P.L. Papini). Critical Exponents (V. Pták). Fourier series and spherical harmonics in convexity (H. Groemer). Zonoids and generalisations (P. Goodey, W. Weil). Baire categories in convexity (P.M. Gruber). Part 5: Stochastic Aspects of Convexity. Integral geometry (R. Schneider, J.A. Wieacker). Stochastic geometry (W. Weil, J.A. Wieacker). Author Index. Subject Index.