ConvexityCUP Archive, 1958 - 141 pages This account of convexity includes the basic properties of convex sets in Euclidean space and their applications, the theory of convex functions and an outline of the results of transformations and combinations of convex sets. It will be useful for those concerned with the many applications of convexity in economics, the theory of games, the theory of functions, topology, geometry and the theory of numbers. |
Table des matières
Preface page vii | 1 |
Intersections closures and interiors of convex sets | 7 |
The dimension of a convex set Barycentric | 13 |
Separation of convex sets and support hyperplanes | 19 |
Duality in Euclidean space | 25 |
Continuous mappings of convex sets Regular | 31 |
The relation of Hellys theorem to Carathéodorys | 39 |
GENERAL PROPERTIES | 45 |
Sets and numbers associated with a convex set | 76 |
Mixed volumes | 82 |
Surface area | 88 |
Central symmetrization | 101 |
The isoperimetric inequality in R2 | 107 |
Plane convex sets | 115 |
SETS OF CONSTANT WIDTH | 122 |
Minimal universal covers | 131 |
APPROXIMATIONS TO CONVEX SETS | 59 |
Approximations by convex polytopes and regular | 67 |
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Expressions et termes fréquents
affine affine transformation applied arcs argument array belong bounded convex set central centre CHAPTER closed bounded convex complete concave condition consider constant width contains continuous converges convex cover convex function convex set COROLLARY corresponding cuts defined definition Denote diameter dimension direction distance dual equal example EXERCISES exists faces fact finite number follows frontier function Further gauge function give given Helly's theorem Hence holds implies inequality interior point intersection least length lies linear manifold meets n-dimensional obtain origin parallel particular perpendicular plane polytope positive number possible problems projection proof properties proved radius regular relation relative interior respectively result segment separates sequence set of constant side Similarly simplex space sphere subset support hyperplane Suppose symmetrization theorem transformation triangle true unit vector vertices write X₁ X₂ y₁ zero