Handbook of Convex Geometry

Handbook of Convex Geometry, Volume A offers a survey of convex geometry and its many ramifications and relations with other areas of mathematics, including convexity, geometric inequalities, and convex sets.

The selection first offers information on the history of convexity, characterizations of convex sets, and mixed volumes. Topics include elementary convexity, equality in the Aleksandrov-Fenchel inequality, mixed surface area measures, characteristic properties of convex sets in analysis and differential geometry, and extensions of the notion of a convex set. The text then reviews the standard isoperimetric theorem and stability of geometric inequalities.

The manuscript takes a look at selected affine isoperimetric inequalities, extremum problems for convex discs and polyhedra, and rigidity. Discussions focus on include infinitesimal and static rigidity related to surfaces, isoperimetric problem for convex polyhedral, bounds for the volume of a convex polyhedron, curvature image inequality, Busemann intersection inequality and its relatives, and Petty projection inequality. The book then tackles geometric algorithms, convexity and discrete optimization, mathematical programming and convex geometry, and the combinatorial aspects of convex polytopes.

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

Inhalt

VOLUME A. Preface. History of Convexity (P.M. Gruber). Part 1: Classical Convexity. Characterizations of convex sets (P. Mani-Levitska). Mixed volumes (J.R. Sangwine-Yager). The standard isoperimetric theorem (G. Talenti). Stability of geometric inequalities (H. Groemer). Selected affine isoperimetric inequalities (E. Lutwak). Extremum problems for convex discs and polyhedra (A. Florian). Rigidity (R. Connelly). Convex surfaces, curvature and surface area measures (R. Schneider). The space of convex bodies (P.M. Gruber). Aspects of approximation of convex bodies (P.M. Gruber). Special convex bodies (E. Heil, H. Martini). Part 2: Combinatorial Aspects of Convexity. Helly, Radon, and Carathéodory type theorems (J. Eckhoff). Problems in discrete and combinatorial geometry (P. Schmitt). Combinatorial aspects of convex polytopes (M.M. Bayer, C.W. Lee). Polyhedral manifolds (U. Brehm, J.M. Wills). Oriented matroids (J. Bokowski). Algebraic geometry and convexity (G. Ewald). Mathematical programming and convex geometry (P. Gritzmann, V. Klee). Convexity and discrete optimization (R.E. Burkard). Geometric algorithms (H. Edelsbrunner). Author Index. Subject Index.