## Convex Polytopes and the Upper Bound Conjecture |

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### Contents

CONVEX POLYTOPES | 39 |

GALE DIAGRAMS AND POLYTOPES WITH | 119 |

THE UPPER BOUND CONJECTURE | 152 |

THE UPPER BOUND CONJECTURE | 169 |

References | 180 |

### Common terms and phrases

aff F affine subspace affine transformation affinely independent basis Q bounded convex set Chapter clearly closed bounded convex closed half-spaces coface combinatorial types combinatorially equivalent completes the proof containing convex combination convex hull convex polytope Corollary cyclic polytope C(v d-dimensional d-polytope d-pyramid d-simplex deduce defined definition Dehn-Sommerville equations denote dimension Euler equation face F face of F facets Figure follows Gale transform Hence hyperplane H implies intersection k-faces k-neighbourly polytope l)-polytope Lemma Let F linear subspace nearest point map neighbourly non-negative non-singular number of faces o e int open half-space orthogonal projection polar set proof of Theorem proper face Proposition prove pyramid Radon's Theorem relative interior result set of points set of vertices simplex simplicial polytopes spherical complexes standard Gale diagram supporting hyperplane Theorem 13 transform of vert Upper Bound Conjecture usual convention vector vert F vertex vertex-figure write