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Combinatorial Convexity

Imre Bárány Rényi Institute of Mathematics, Budapest, Hungary and University College London, London, United Kingdom
Available Formats:
Softcover ISBN: 978-1-4704-6709-8
Product Code: ULECT/77
List Price: $55.00 MAA Member Price:$49.50
AMS Member Price: $44.00 Electronic ISBN: 978-1-4704-6768-5 Product Code: ULECT/77.E List Price:$55.00
MAA Member Price: $49.50 AMS Member Price:$44.00
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List Price: $82.50 MAA Member Price:$74.25
AMS Member Price: $66.00 Click above image for expanded view Combinatorial Convexity Imre Bárány Rényi Institute of Mathematics, Budapest, Hungary and University College London, London, United Kingdom Available Formats:  Softcover ISBN: 978-1-4704-6709-8 Product Code: ULECT/77  List Price:$55.00 MAA Member Price: $49.50 AMS Member Price:$44.00
 Electronic ISBN: 978-1-4704-6768-5 Product Code: ULECT/77.E
 List Price: $55.00 MAA Member Price:$49.50 AMS Member Price: $44.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$82.50 MAA Member Price: $74.25 AMS Member Price:$66.00
• Book Details

University Lecture Series
Volume: 772021; 148 pp
MSC: Primary 52; 05;

This book is about the combinatorial properties of convex sets, families of convex sets in finite dimensional Euclidean spaces, and finite points sets related to convexity. This area is classic, with theorems of Helly, Carathéodory, and Radon that go back more than a hundred years. At the same time, it is a modern and active field of research with recent results like Tverberg's theorem, the colourful versions of Helly and Carathéodory, and the $(p, q)$ theorem of Alon and Kleitman. As the title indicates, the topic is convexity and geometry, and is close to discrete mathematics. The questions considered are frequently of a combinatorial nature, and the proofs use ideas from geometry and are often combined with graph and hypergraph theory.

The book is intended for students (graduate and undergraduate alike), but postdocs and research mathematicians will also find it useful. It can be used as a textbook with short chapters, each suitable for a one- or two-hour lecture. Not much background is needed: basic linear algebra and elements of (hyper)graph theory as well as some mathematical maturity should suffice.

An instructor's manual for this title is available electronically to those instructors who have adopted the textbook for classroom use. Please send email to textbooks@ams.org for more information.

Undergraduate and graduate students and researchers interested in combinatorial properties of convexity and convex sets.

• Chapters
• Basic concepts
• Carathéodory’s theorem
• Tverberg’s theorem
• General position
• Helly’s theorem
• Applications of Helly’s theorem
• Fractional Helly
• Colourful Carathéodory
• Colourful Carathéodory again
• Colourful Helly
• Tverberg’s theorem again
• Colourful Tverberg theorem
• Sarkaria and Kirchberger generalized
• The Erdős-Szekers theorem
• The same type lemma
• Better bound for the Erdős-Szekeres number
• Covering number, planar case
• The stretched grid
• Covering number, general case
• Upper bound on the covering number
• The point selection theorem
• Homogeneous selection
• Missing few simplices
• Weak $\varepsilon$-nets
• Lower bound on the size of weak $\varepsilon$-nets
• The $(p,q)$ theorem
• The colourful $(p,q)$ theorem
• $d$-intervals
• Halving lines, havling planes
• Convex lattice sets
• Fractional Helly for convex lattice sets

• Reviews

• It is a real gift for students and the much larger readership if they can learn firsthand from an active researcher in a subject. Imre Bárány is one of them; in particular, his work has been a driving force behind the recent progress of combinatorial convexity. His book of the highest standard can be used as a textbook for graduate or undergraduate courses. The short chapters are suitable for one- or two-hour lectures. At the end of each chapter, various exercises complete the material and help deepen understanding. Basic linear algebra, linear programming, and some experience in graph and hypergraph theory, that is, certain mathematical maturity, are expected from the reader.

Jeno Lehel (University of Memphis), MathSciNet
• This is an elegant, well written, concise treatment of an attractive and active subject, written by an expert who has made important contributions to the area himself. I am sure this will be a successful textbook.

Noga Alon, Princeton University and Tel Aviv University
• I think this book is a gem.

János Pach, Rényi Institute of Mathematics, Budapest
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 772021; 148 pp
MSC: Primary 52; 05;

This book is about the combinatorial properties of convex sets, families of convex sets in finite dimensional Euclidean spaces, and finite points sets related to convexity. This area is classic, with theorems of Helly, Carathéodory, and Radon that go back more than a hundred years. At the same time, it is a modern and active field of research with recent results like Tverberg's theorem, the colourful versions of Helly and Carathéodory, and the $(p, q)$ theorem of Alon and Kleitman. As the title indicates, the topic is convexity and geometry, and is close to discrete mathematics. The questions considered are frequently of a combinatorial nature, and the proofs use ideas from geometry and are often combined with graph and hypergraph theory.

The book is intended for students (graduate and undergraduate alike), but postdocs and research mathematicians will also find it useful. It can be used as a textbook with short chapters, each suitable for a one- or two-hour lecture. Not much background is needed: basic linear algebra and elements of (hyper)graph theory as well as some mathematical maturity should suffice.

An instructor's manual for this title is available electronically to those instructors who have adopted the textbook for classroom use. Please send email to textbooks@ams.org for more information.

Undergraduate and graduate students and researchers interested in combinatorial properties of convexity and convex sets.

• Chapters
• Basic concepts
• Carathéodory’s theorem
• Tverberg’s theorem
• General position
• Helly’s theorem
• Applications of Helly’s theorem
• Fractional Helly
• Colourful Carathéodory
• Colourful Carathéodory again
• Colourful Helly
• Tverberg’s theorem again
• Colourful Tverberg theorem
• Sarkaria and Kirchberger generalized
• The Erdős-Szekers theorem
• The same type lemma
• Better bound for the Erdős-Szekeres number
• Covering number, planar case
• The stretched grid
• Covering number, general case
• Upper bound on the covering number
• The point selection theorem
• Homogeneous selection
• Missing few simplices
• Weak $\varepsilon$-nets
• Lower bound on the size of weak $\varepsilon$-nets
• The $(p,q)$ theorem
• The colourful $(p,q)$ theorem
• $d$-intervals
• Halving lines, havling planes
• Convex lattice sets
• Fractional Helly for convex lattice sets
• It is a real gift for students and the much larger readership if they can learn firsthand from an active researcher in a subject. Imre Bárány is one of them; in particular, his work has been a driving force behind the recent progress of combinatorial convexity. His book of the highest standard can be used as a textbook for graduate or undergraduate courses. The short chapters are suitable for one- or two-hour lectures. At the end of each chapter, various exercises complete the material and help deepen understanding. Basic linear algebra, linear programming, and some experience in graph and hypergraph theory, that is, certain mathematical maturity, are expected from the reader.

Jeno Lehel (University of Memphis), MathSciNet
• This is an elegant, well written, concise treatment of an attractive and active subject, written by an expert who has made important contributions to the area himself. I am sure this will be a successful textbook.

Noga Alon, Princeton University and Tel Aviv University
• I think this book is a gem.

János Pach, Rényi Institute of Mathematics, Budapest
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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