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Geometry and Convexity: A Study in Mathematical Methods

Dover Publications
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This is a reprint of the edition published by John Wiley and Sons in 1979. Reading and working through the book was simply fun. I liked the flow of the book: the topics seemed to follow naturally and the exposition moved along at pace that kept my interest.

The book’s domain is in something that students should grasp, Euclidean n-space. But the material is developed in a way that could easily be generalized to arbitrary spaces. The book starts with a chapter on topology and metrics, then applies this to Euclidean n-space. There follow a couple of chapters on convex sets, with notions such as width of a set, and set dimension explored. The last two chapters tackle notions such as convex spans, convex hulls, extreme points, linear combination of sets, and some material on the Haudorff metric.

The material in the book is not all straightforward. Mathematical challenges such as existence proofs, uniqueness proofs, and deeper philosophical questions are tackled surprisingly often. For example, near the end of the book in section on existence proofs, it reads, “ …simply to emphasize how subtle and difficult existence questions can be.”

The book for the most part is beautifully self-contained: some background in linear algebra, topology, abstract algebra while very helpful, is not absolutely necessary. There is a subject index; several times, when I forgot the definition of something, the index came to my rescue. The book contains a reasonable number of exercises, most of which were interesting little problems in their own right. I found the exercises not so difficult as to be frustrating but not so trivial as to be boring.

Usually when one goes through a book of this size, one finds typesetting errors. I never came across any. So if there are mistakes there can’t be very many of them, another big plus for undergraduate students.

Overall, I think the book would make a highly successful text for a mathematics course on convexity, or even as a helper text in a class on mathematical proofs or topology or linear algebra. Many of the mathematical ideas are broadly used throughout mathematics and this book captures the mathematical thinking nicely, with a strong emphasis on mathematical rigor and precision.

If you enjoy doing mathematics or you are looking for a book with some foundational mathematical material for undergraduates, I recommend this book highly. The price is very reasonable, and even if the binding is a simple glue binding it seemed to standup to my carrying the book around to a pile of appointments and meetings. Since I could follow the book using fragmented time, my feeling is that if students put forward any reasonable effort they will thrive on the clear expositions and pick up valuable proof and mathematical skills in the process.

Collin Carbno is a specialist in process improvement and methodology. He holds a Master’s of Science Degree in theoretical physics and completed course work for Ph.D. in theoretical physics (relativistic rotating stars) in 1979 at the University of Regina. He has been employed for nearly 30 years in various IT and process work at Saskatchewan Telecommunications and currently holds a Professional Physics Designation from the Canadian Association of Physicists, and the Information System Professional designation from the Canadian Information Process Society.

Date Received: 
Sunday, August 30, 2009
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Paul J. Kelly and Max L. Weiss
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Collin Carbno
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Tuesday, July 27, 2010