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**Author(s):** Alexander Kharazishvili

**Publisher:** CRC Press

**Year:** 2024

**Language:** English

Cover

Half Title

Title Page

Copyright Page

Contents

Preface

CHAPTER 1: The index of an isometric embedding

CHAPTER 2: Maximal ot-subsets of the Euclidean plane

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

CHAPTER 4: Isosceles triangles and it-sets in Euclidean space

CHAPTER 5: Some geometric consequences of Ramsey’s combinatorial theorem

CHAPTER 6: Convexly independent subsets of infinite sets of points

CHAPTER 7: Homogeneous coverings of the Euclidean plane

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

CHAPTER 9: Chromatic numbers of graphs associated with point sets in Euclidean space

CHAPTER 10: The Szemerédi–Trotter theorem and its applications

CHAPTER 11: Minkowski’s theorem, number theory, and nonmeasurable sets

CHAPTER 12: Tarski’s plank problem

CHAPTER 13: Borsuk’s conjecture

CHAPTER 14: Piecewise affine approximations of continuous functions of several variables and Caratheodory–Gale polyhedra

CHAPTER 15: Dissecting a square into triangles of equal areas

CHAPTER 16: Geometric realizations of finite and infinite families of sets

CHAPTER 17: A geometric form of the Axiom of Choice

APPENDIX 1: Convex sets in real vector spaces

APPENDIX 2: Real-valued convex functions

APPENDIX 3: The Principle of Inclusion and Exclusion

APPENDIX 4: The Erdös–Mordell inequality

APPENDIX 5: Some facts from graph theory

BIBLIOGRAPHY

INDEX

Half Title

Title Page

Copyright Page

Contents

Preface

CHAPTER 1: The index of an isometric embedding

CHAPTER 2: Maximal ot-subsets of the Euclidean plane

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

CHAPTER 4: Isosceles triangles and it-sets in Euclidean space

CHAPTER 5: Some geometric consequences of Ramsey’s combinatorial theorem

CHAPTER 6: Convexly independent subsets of infinite sets of points

CHAPTER 7: Homogeneous coverings of the Euclidean plane

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

CHAPTER 9: Chromatic numbers of graphs associated with point sets in Euclidean space

CHAPTER 10: The Szemerédi–Trotter theorem and its applications

CHAPTER 11: Minkowski’s theorem, number theory, and nonmeasurable sets

CHAPTER 12: Tarski’s plank problem

CHAPTER 13: Borsuk’s conjecture

CHAPTER 14: Piecewise affine approximations of continuous functions of several variables and Caratheodory–Gale polyhedra

CHAPTER 15: Dissecting a square into triangles of equal areas

CHAPTER 16: Geometric realizations of finite and infinite families of sets

CHAPTER 17: A geometric form of the Axiom of Choice

APPENDIX 1: Convex sets in real vector spaces

APPENDIX 2: Real-valued convex functions

APPENDIX 3: The Principle of Inclusion and Exclusion

APPENDIX 4: The Erdös–Mordell inequality

APPENDIX 5: Some facts from graph theory

BIBLIOGRAPHY

INDEX