Surveys on Discrete and Computational Geometry : Twenty Years Later.

Intro -- Contents -- Preface -- Musings on discrete geometry and "20 years of Discrete &amp -- Computational Geometry -- State of the union (of geometric objects) -- Metric graph theory and geometry: a survey -- Extremal problems for convex lattice polytopes: a survey -- On simple arrangements of lines and pseudo-lines in P2 and R2 with the maximum number of triangles -- The computational complexity of convex bodies -- Algorithmic semi-algebraic geometry and topology - recent progress and open problems -- 1. Introduction -- 2. Semi-algebraic Geometry: Background -- 3. Recent Algorithmic Results -- 4. Algorithmic Preliminaries -- 5. Topological Preliminaries -- 6. Algorithms for Computing the First Few Betti Numbers -- 7. The Quadratic Case -- 8. Betti Numbers of Arrangements -- 9. Open Problems -- Acknowledgment -- References -- Expansive motions -- All polygons flip finitely … right? -- Persistent homology-a survey -- Recent progress on line transversals to families of translated ovals -- An improved, simple construction of many halving edges -- Unfolding orthogonal polyhedra -- The discharging method in combinatorial geometry and the Pach-Sharir conjecture -- Pseudo-triangulations-a survey -- 1. Introduction -- 2. Basic Properties of Pseudo-Triangulations -- 3. The Set of all Pseudo-Triangulations -- 4. 3D Liftings and Locally Convex Functions -- 5. Self-Stresses, Reciprocal Diagrams, and the Maxwell-Cremona Correspondence -- 6. Pseudo-Triangulations and Rigidity -- 7. Planar Rigid Graphs are Pseudo-Triangulations -- 8. Polytopes of Pseudo-Triangulations -- 9. Applications of Pseudo-Triangulations -- References -- Line problems in nonlinear computational geometry -- On empty hexagons -- k-sets and k-facets -- 1. Introduction -- 2. Preliminaries -- 3. Random Sampling -- 4. Special Point Sets -- 5. Lower Bounds.

6. Upper Bounds for Halving Facets in All Dimensions -- 7. Crossings in Dimension 2. -- 8. Improvements in Three And Four Dimensions -- 9. Convex Quadrilaterals -- 10. Connections to the Combinatorial Theory of Convex Polytopes -- References -- An Erdős-Szekeres type problem for interior points -- The kissing number, blocking number and covering number of a convex body -- Open problems.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.