Results and Problems in Combinatorial Geometry by V. G. Boltianskii – A Cambridge University Press Classic on Borsuk's P

Introduction

Combinatorial geometry is a fascinating branch of mathematics that sits at the intersection of geometry and combinatorics. Results and Problems in Combinatorial Geometry by V. G. Boltianskii offers a rare and accessible entry point into this field, especially for Indian students and enthusiasts who wish to explore advanced geometric ideas without requiring a university-level background. Published by Cambridge University Press, this hardcover volume is a treasure trove of elegant problems and profound results, making it an ideal companion for self-study or classroom enrichment.

Book Overview

This compact yet powerful book focuses on three classic problems in combinatorial geometry: Borsuk's partition problem, the covering of convex bodies by smaller homothetic bodies, and the illumination problem. The author demonstrates how these seemingly distinct questions are deeply interconnected, revealing the unity of geometric reasoning. The text is written in an elementary style, assuming only high-school mathematics and a genuine curiosity for geometry. Most discussions are confined to two- and three-dimensional Euclidean space, though occasional generalizations are provided. The book also includes a rich collection of unsolved and partially solved problems that even a sixth-form student can understand and attempt to tackle.

Key Highlights

• Three Core Problems: In-depth exploration of Borsuk's partition, homothetic covering, and illumination problems.
• Elementary Approach: Requires only high-school mathematics, making advanced concepts accessible to Indian students.
• Interconnected Themes: Shows how different geometric problems are linked, fostering a holistic understanding.
• Unsolved Problems: A dedicated section with open problems that challenge and inspire young mathematicians.
• Practical Applications: Discusses relevance to mathematical programming, operations research, and theoretical computer science.

Inside the Book

The book is structured to guide readers from foundational concepts to cutting-edge questions. Early chapters introduce convex bodies, partitions, and homothety, building a solid base. The middle sections delve into the three main problems, presenting proofs and counterexamples with clarity. The final part is a goldmine of unsolved problems, each carefully described so that a motivated student can begin exploring. Diagrams and step-by-step reasoning accompany the text, ensuring that visual learners can follow along easily.

Key Topics

• Borsuk's Partition Problem: Dividing a set into parts of smaller diameter.
• Covering Convex Bodies: Using smaller homothetic copies to cover a given shape.
• Illumination Problem: Determining the minimum number of directions needed to illuminate a convex body.
• Convexity and Geometry: Basic properties of convex sets in two and three dimensions.
• Open Research Questions: Problems that remain unsolved, offering opportunities for original work.

Reader Benefits

• Builds Mathematical Maturity: Encourages logical thinking and problem-solving skills.
• Bridges School and College Math: Prepares Indian students for advanced studies in mathematics and computer science.
• Self-Study Friendly: Clear explanations and exercises make it suitable for independent learners.
• Inspires Research: The unsolved problems can spark a lifelong interest in mathematical discovery.
• Compact and Focused: A short book that delivers deep insights without overwhelming the reader.

Learning Outcomes

By the end of this book, readers will be able to understand and explain the three classic problems of combinatorial geometry. They will gain proficiency in constructing geometric arguments, recognizing connections between different areas of mathematics, and approaching unsolved problems with confidence. The book also equips students with the vocabulary and conceptual tools needed to explore further literature in geometry and discrete mathematics.

Who Should Read

• High School Students: Especially those preparing for Olympiads or competitive exams like JEE or ISI.
• Undergraduate Beginners: First-year college students in mathematics, physics, or computer science.
• Teachers and Educators: Looking for enrichment material for bright students.
• Math Enthusiasts: Anyone with a love for geometry and puzzles.
• Researchers: Professionals seeking a concise reference on these classical problems.

About the Author

V. G. Boltianskii was a distinguished Soviet mathematician known for his contributions to geometry, topology, and combinatorial mathematics. His works are celebrated for their clarity and pedagogical value, making advanced topics accessible to a wide audience. This book reflects his commitment to sharing the beauty of geometry with students and non-specialists alike.

About the Publisher

Cambridge University Press is one of the world's oldest and most respected academic publishers. With a legacy of excellence spanning over four centuries, Cambridge ensures that each title meets the highest standards of scholarship and production. This hardcover edition is printed on quality paper and bound to last, making it a valuable addition to any personal or institutional library.

Conclusion

Results and Problems in Combinatorial Geometry is a must-have for anyone who wants to explore the elegance of geometry beyond the classroom. Its blend of classical results, modern applications, and open problems offers a unique intellectual adventure. Whether you are a student in Mumbai, a teacher in Delhi, or a math lover in Bengaluru, this book will challenge and delight you. Add it to your collection today from Bookshops.in and experience the thrill of combinatorial geometry.