Convexity

Title ......Page 1
Contents ......Page 3
Preface ......Page 5
Introduction ......Page 6
On systems of linear inequalities in Hermitian matrix variables. R.Bellman and Ky Fan ......Page 13
Minimum area of a set of constant width. A.S.Besicovitch ......Page 25
On semicircles inscribed into sets of constant width. A.S.Besicovitch ......Page 27
A cage to hold a unit-sphere. A.S.Besicovitch ......Page 31
On singular points of convex surfaces. A.S.Besicovitch ......Page 33
On the set of directions of linear segments on a convex surface. A.S.Besicovitch ......Page 36
The support functionals of a convex set. R.Phelps ......Page 39
Topological classification of convex sets. H.Corson and V.Klee ......Page 49
An upper bound for the number of equal nonoverlapping spheres that can touch another of the same size. H.Coxeter ......Page 65
Rotundity. D.F.Cudia ......Page 85
A characterization of the circle. L.Danzer ......Page 111
Helly's theorem and its relatives. Danzer, Grunbaum and Klee ......Page 113
An extremal problem for plane convex curves. C.Davis ......Page 193
Notions generalizing convexity for functions defined on spaces of matrices. C.Davis ......Page 199
Some near-sphericity results. A.Dvoretzky ......Page 215
On the Krein-Milman theorem. Ky Fan ......Page 223
On Lipschitzian mappings of convex bodies. D.Gale ......Page 233
Neighborly and cyclic polytopes. D.Gale ......Page 237
Measures of symmetry for convex sets. B.Grunbaum ......Page 245
Borsuk's problem and related questions. B.Gbunbaum ......Page 283
On polyhedral graphs. B.Gbunbaum and T.S.Motzkin ......Page 297
Convex curves of constant Minkowski breadth. P.C.Hammer ......Page 303
Semispaces and the topology of convexity. P.C.Hammer ......Page 317
On simple linear programming problems. A.J.Hoffman ......Page 329
Total positivity and convexity preserving transformations. S.Karlin ......Page 341
Infinite-dimensional intersection theorems. V.Klee ......Page 361
Endovectors. T.Motzkin ......Page 373
Representation of points of a set as linear combinations of boundary points. T.Motzkin, E.Straus ......Page 401
Support cones and their generalizations. R.Phelps ......Page 405
Convex spaces associated with a family of linear inequalities. H.Poritsky ......Page 415
A combinatorial lemma on the existence of convex means and its application to weak compactness. V.Ptak ......Page 449
Convex cones and spectral theory. H.Schaefer ......Page 463
The dual cone and Helly type theorems. F.Valentine ......Page 485
Unsolved Problems ......Page 507
Index of Unsolved Problems ......Page 513
Author Index ......Page 515
Subject Index ......Page 521