New Trends in Discrete and Computational Geometry by Janos Pach (PDF)

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I. Combinatorics and Algorithms of Arrangements.- 1. Introduction.- 2. Arrangements of Curves in the Plane.- 3. Lower Envelopes and Davenport-Schinzel Sequences.- 3.1 A Few Initial Applications.- 3.2 Applications of Lower Envelopes and Davenport-Schinzel Sequences.- 4. Faces in Arrangements.- 4.1 The Case of a Single Face.- 4.2 Many Faces in Arrangements and Related Problems.- 4.3 Partitioning Arrangements.- 5. Arrangements in Higher Dimensions.- 5.1 Lower Envelopes of Bivariate Functions and Their Applications.- 5.2 The Complexity of a Single Cell and Related Problems.- 6. Summary.- References.- II. Backwards Analysis of Randomized Geometric Algorithms.- 1. Introduction.- 2. Delaunay Triangulations of Convex Polygons.- 3. Intersecting Line Segments.- 4. Constructing Planar Convex Hulls.- 5. Backwards Analysis of QUICKSORT.- 6. A Bad Example.- 7. Linear Programming for Small Dimension.- 8. Welzl’s Minidisk Algorithm.- 9. Clarkson’s Backwards Analysis of the Conflict Graph Based on the Convex Hull Algorithm.- 10. Odds and Ends.- References.- III. Epsilon-Nets and Computational Geometry.- 1. Range Spaces and ?-Nets.- 2. Geometric Range Spaces.- 3. A Sample of Applications.- 4. Removing Logarithms.- 5. Removing the Randomization.- References.- IV. Complexity of Polytope Volume Computation.- 1. Jumps of the Derivatives.- 2. Exact Volume Computation is Hard.- 3. Volume Approximation.- References.- V. Allowable Sequences and Order Types in Discrete and Computational Geometry.- 1. Introduction.- 2. Combinatorial Types of Configurations in the Plane and Allowable Sequences.- 3. Arrangements of Lines and Pseudolines.- 4. Applications of Allowable Sequences.- 4.1 Stretchability of Pseudoline Arrangements.- 4.2 The Directions Problem.- 4.3 k-Sets.- 5. Order Types of Points in Rd and “Geometric Sorting”.- 6. The Number of Order Types in Rd.- 7. Isotopy and Realizability Questions.- 8. Lattice Realization of Order Types and the Problem of Robustness in Computational Geometry.- References.- VI. Hyperplane Approximation and Related Topics.- 1. Introduction.- 2. MINSUM Problem: Orthogonal L1-Fit.- 2.1 Basic Facts.- 2.2 General Case: O(nd log n) Time and Linear Space Algorithm.- 2.3 General Case: O(nd) Time.- 2.4 Restrictions and Extensions.- 3. MINSUM Problem: Vertical L1-Fit.- 4. MINMAX Problem: Orthogonal L?-Fit.- 4.1 Unweighted Case.- 4.2 Weighted Case.- 4.3 MINMAX Problem: k-Flats.- 5. MINMAX Problem: Vertical L?-Fit.- 6. Related Issues.- 6.1 Stabbing Lines/Hyperplanes.- 6.2 k-Hyperplane Problems.- References.- VII. Geometric Transversal Theory.- 1. Introduction.- 2. Hadwiger-Type Theorems.- 3. The Combinatorial Complexity of the Space of Transversals.- 4. Translates of a Convex Set.- 5. Transversal Algorithms.- 6. Other Directions.- References.- VIII. Hadwiger-Levi’s Covering Problem Revisited.- 0. Introduction.- 1. On I0(K) and I?(K).- 2. On Il(K) and k-fold Illumination.- 3. Some Simple Remarks on H(B).- 4. On Convex Bodies with Finitely Many Corner Points.- 5. Solution of Hadwiger-Levi’s Covering Problem for Convex Polyhedra with Affine Symmetry.- References.- IX. Geometric and Combinatorial Applications of Borsuk’s Theorem.- 1. Introduction.- 2. Van Kampen-Flores Type Results.- 3. The Ham-Sandwich Theorem.- 4. Centrally Symmetric Polytopes.- 5. Kneser’s Conjecture.- 6. Sphere Coverings.- References.- X. Recent Results in the Theory of Packing and Covering.- 1. Introduction.- 2. Preliminaries and Basic Concepts.- 3. A Review of Some Classical Results in the Plane.- 4. Economical Packing in and Covering of the Plane.- 5. Multiple Packing and Covering.- 6. Some Computational Aspects of Packing and Covering.- 7. Restrictions on the Number of Neighbors in a Packing.- 8. Selected Topics in 3 Dimensions.- References.- XI. Recent Developments in Combinatorial Geometry.- 1. The Distribution of Distances.- 2. Graph Dimensions.- 3. Geometric Graphs.- 4. Arrangements of Lines in Space.- References.- XII. Set Theoretic Constructions in Eucl

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