1st Edition

Introduction to Combinatorial Methods in Geometry

396 Pages

by Chapman & Hall

396 Pages

by Chapman & Hall

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Description

This book offers an introduction to some combinatorial (also, set-theoretical) approaches and methods in geometry of the Euclidean space R m . The topics discussed in the manuscript are due to the field of combinatorial and convex geometry. The author’s primary intention is to discuss those themes of Euclidean geometry which might be of interest to a sufficiently wide audience of potential... Read more

1. The index of an isometric embedding

2. Maximal ot-subsets of the Euclidean plane

3. The cardinalities of at-sets in a real Hilbert space

4. Isosceles triangles and it-sets in Euclidean space

5. Some geometric consequences of Ramsey’s combinatorial theorem

6. Convexly independent subsets of infinite sets of points

7. Homogeneous coverings of the Euclidean plane

8. Three-colorings of the Euclidean plane and associated triangles of a prescribed type

9. Chromatic numbers of graphs associated with point systems in Euclidean space

10. The Szemeredi–Trotter theorem

11. Minkowski’s theorem, number theory, and nonmeasurable sets

12. Tarski’s plank problem

13. Borsuk’s conjecture

14. Piecewise affine approximations of continuous functions of several variables and Caratheodory–Gale polyhedral

15. Dissecting a square into triangles of equal areas

16. Geometric realizations of finite and infinite families of sets