This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems. Combinatorial investigations play an important role in this study.

Algorithms in Combinatorial Geometry

1 Convex Sets.- 1. The Affine Structure of ?d.- 2. Convex Sets.- 3. The Relative Interior of a Convex Set.- 4. Supporting Hyperplanes and Halfspaces.- 5. The Facial Structure of a Closed Convex Set.- 6. Polarity.- 2 Convex Polytopes.- 7. Polytopes.- 8. Polyhedral Sets.- 9. Polarity of Polytopes and Polyhedral Sets.- 10. Equivalence and Duality of Polytopes.- 11. Vertex-Figures.- 12. Simple and Simplicial Polytopes.- 13. Cyclic Polytopes.- 14. Neighbourly Polytopes.- 15. The Graph of a Polytope.- 3 Combinatorial Theory of Convex Polytopes.- 16. Euler s Relation.- 17. The Dehn-Sommerville Relations.- 18. The Upper Bound Theorem.- 19. The Lower Bound Theorem.- 20. McMullen s Conditions.- Appendix 1 Lattices.- Appendix 2 Graphs.- Appendix 3 Combinatorial Identities.- Bibliographical Comments.- List of Symbols.

https://www.fmf.uni-lj.si/~lavric/brondsted.pdf

An Introduction to Convex Polytopes

We regard a graph G as a set { 1,...,v) together with a nonempty set E of two-element subsets of (1, . . ., v). Letp = (Pi, . . . &) be an element of RIv representing v points in RI. Consider the figure G(p) in RI consisting of the line segments [pi, pj in RI for {i,j) E E. The figure G (p) is said to be rigid in RI if every continuous path in RI', beginning atp and preserving the edge lengths of G(p), terminates at a point q E RIV which is the image (Tp1, . . ., Tpv) of p under an isometry T of RW. Otherwise, G (p) is flexible in RW. Our main result establishes a formula for determining whether G (p) is rigid in RI for almost all locations p of the vertices. Applications of the formula are made to complete graphs, planar graphs, convex polyhedra in R3, and other related matters.

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The rigidity of graphs

Introduction Polytopes Topology and combinatorics of simplicial complexes Commutative and homological algebra of simplicial complexes Cubical complexes Toric and quasitoric manifolds Moment-angle complexes Cohomology of moment-angle complexes and combinatorics of triangulated manifolds Cohomology rings of subspace arrangement complements Bibliography Index.

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Torus Actions and their Applications in Topology and Combinatorics

© Séminaire analyse fonctionnelle (dit \"Maurey-Schwartz\") (École Polytechnique), 1975-1976, tous droits réservés. L’accès aux archives du séminaire d’analyse fonctionnelle implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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The dimension of almost spherical sections of convex bodies

Ad-polytope is ad-dimensional set that is the convex hull of a finite number of points. Ad-polytope is simple provided each vertex meets exactlyd edges. It has been conjectured that for simple polytopes {fx121-1} wheref
 i is the number ofi-dimensional faces of the polytope. In this paper we show that inequality (i) holds for all simple polytopes.

The minimum number of vertices of a simple polytope

Generating the triangulations of the projective plane

Two basic theorems about the graphs of convex polytopes are that the graph of ad-polytope isd-connected and that it contains a refinement of the complete graph ond+1 vertices. We obtain generalizations of these theorems, and others, for manifolds. We also supply some details for a proof of the lower bound inequality for manifolds.

Graph theorems for manifolds

: Two relatively simple proofs for the famous theorem of Steinitz on the characterization of graphs of 3-polytopes are presented, together with some new results on the structure of 3-connected graphs. (Author)

On Steinitz's theorem concerning convex 3-polytopes and on some properties of planar graphs

A triangulation of a 2-manifoldM is said to be minimal provided one cannot produce a triangulation ofM with fewer vertices by shrinking an edge. In this paper we prove that all 2-manifolds have finitely many minimal triangulations. It follows that all triangulations of a given 2-manifold can be generated from the minimal triangulations by a process called vertex splitting.

David Barnette

Papers

The minimum number of vertices of a simple polytope

Generating the triangulations of the projective plane

Graph theorems for manifolds

On Steinitz's theorem concerning convex 3-polytopes and on some properties of planar graphs

All 2-manifolds have finitely many minimal triangulations