## ConvexityThis 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. |

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### Contents

Convex polytopes | 29 |

The relation of Hellys theorem to Caratheodorys | 39 |

general properties | 45 |

Differential conditions for convexity | 51 |

approximations to convex sets | 59 |

Approximations by convex polytopes and regular | 67 |

Sets and numbers associated with a convex set | 76 |

Mixed volumes | 82 |

Surface area | 88 |

the Minkowski | 96 |

some special problems | 103 |

Relations between the inradius circ urn radius | 111 |

sets of constant width | 122 |

Minimal universal covers | 131 |

139 | |

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### Common terms and phrases

affine transformation belong Blaschke selection theorem Blaschke-Lebesgue theorem bounded convex set bounded set Caratheodory's theorem central convex sets Chapter circumsphere class of convex closed bounded convex closed convex set concave array concave function consider constant width contains convex cover convex function convex polytopes convex subset Corollary defined definition Denote diameter dimension dimensional dual dual space equal Euclidean space extreme points extreme support hyperplanes finite number follows frontier point gauge function Helly's theorem Hence induction inequality insphere integer interior point intersection lemma linear manifold minimal width mixed volumes n-dimensional obtain origin parallelepipeds perpendicular plane convex set points xi positive number problems projection proof properties of convex real numbers relative interior result Reuleaux polygon Reuleaux triangle segment sequence set of constant set of points sets Xi simplex support function Suppose theorem is proved universal cover vertices Xi and X2 xn+1 xv x2 zero