Softcover ISBN:  9781470467098 
Product Code:  ULECT/77 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $44.00 
Electronic ISBN:  9781470467685 
Product Code:  ULECT/77.E 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $44.00 

Book DetailsUniversity Lecture SeriesVolume: 77; 2021; 148 ppMSC: 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 twohour 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.ReadershipUndergraduate and graduate students and researchers interested in combinatorial properties of convexity and convex sets.

Table of Contents

Chapters

Basic concepts

Carathéodory’s theorem

Radon’s theorem

Topological Radon

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 ErdősSzekers theorem

The same type lemma

Better bound for the ErdősSzekeres 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


Additional Material

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 twohour 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


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 Book Details
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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 twohour 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

Radon’s theorem

Topological Radon

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 ErdősSzekers theorem

The same type lemma

Better bound for the ErdősSzekeres 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 twohour 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