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When Less in More: Visualizing Basic Inequalities

Claudi Alsina and Roger B. Nelsen
Mathematical Association of America
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions 36
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
John D. Cook
, on

You can't judge a book by its cover, but a beautiful cover does make you want to look inside. In the case of When Less is More: Visualizing Basic Inequalities by Claudi Alsina and Roger B. Nelsen, appearances are a good guide. The book is beautiful inside and out: nice cover art and filled with uncommonly elegant ideas.

The proofs in When Less is More are in the spirit of proofs without words, though most require at least a few words. The first inequalities presented in the book, such as the inequalities between the harmonic, geometric, and arithmetic mean, are familiar from analysis, but are given geometric proofs. The second and largest set of inequalities are geometric both in their statements and in their proofs. Toward the end of the book some inequalities are more analytical in their statements as well as their proofs.

When Less is More would be a handy reference for someone wanting to spice up a mathematics class with a few visual proofs. For example, the Cauchy-Schwarz inequality is presented as part of many different courses. An instructor could present any of the four visual proofs of this inequality given in the book. As George Pólya once said, "It is better to solve one problem five different ways, than to solve five problems one way." (Pólya was one of three authors of the best-known book on inequalities, simply titled Inequalities.)

The book would also be useful as part of a course on inequalities. (Inequalities are important across mathematics, and yet institutions seldom offer a course that focuses on inequalities. Perhaps that should change.) Such a course might be built around J. Michael Steele's excellent book The Cauchy-Schwarz Master Class with When Less is More used for supplemental material. One could use When Less is More as the primary text, but then the course would be more specifically about geometric inequalities since around half the book is devoted to inequalities stated as problems in geometry. Because many of the topics can be developed independently, one could easily have a seminar on inequalities and not be concerned by sporadic attendance.

If someone were to offer a course in the aesthetics of proofs, When Less is More would be an appropriate textbook, alongside Proofs from THE BOOK by Martin Aigner and Günter M. Ziegler. (The premise of "THE BOOK" is a book that Paul Erdős imagined God kept, recording the most beautiful proof of each theorem.)

A large portion of When Less is More could be understood by someone with only pre-calculus mathematics. Indeed, material from the book could be used to challenge clever high school students who have taken algebra and geometry but who have not yet had calculus.

On the other hand, much of the material would be best appreciated by someone who has seen enough advanced mathematics to understand how inequalities are applied. And while much of the material is elementary, the book offers challenges and surprises even for more sophisticated mathematicians. For example, consider the following problem proposed by Paul Erdős in 1935, now known as the Erdős-Mordell theorem.

From a point O inside a given triangle ABC the perpendiculars OP, OQ, OR are drawn to its sides. Prove that |OA| + |OB| + |OC| ≤ 2(|OP| + |OQ| + |OR|).

This theorem has an elementary statement, but according to When Less is More, the first proof appeared in 1937 and the first "simple and elementary" proof did not appear until 1956.

When Less is More is a pleasure to read.

John D. Cook is a research statistician at M. D. Anderson Cancer Center and blogs daily at The Endeavour.

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