Geometric Inequalities

This is an interesting example of what can be done at the high-school level with almost no prerequisites. It can be studied after an introduction to algebra and geometry. It is primarily a proofs book, but also includes much discussion of how you can discover things. By “geometric inequalities” the author means inequalities that have a geometric interpretation, and most of the discussion is also based on synthetic geometry.

The book begins with a chapter on the arithmetic mean–geometric mean inequality. This is the one part of the book that is primarily algebraic rather than geometric, but this does give a good opportunity for a careful look at the number line, the concept of inequalities, and the algebraic rules for manipulating and proving inequalities.

The center of the book is the isoperimetric problem: what plane figure of a given perimeter has the largest area? The problem is tackled “in the spirit of Steiner” (p. 31), that is, from a synthetic geometry viewpoint rather than an algebraic or analytic one, and works up gradually from solving the problem for simpler polygons to the general case. Steiner assumed the existence of a figure with maximal area and was later criticized for this. The book devotes several pages to understanding why existence really is an issue that has to be dealt with, and considers several other problems where there is indeed not a maximal figure.

The isoperimetric chapter is followed by a chapter on further uses of reflection and symmetry in optimization. This is a more miscellaneous chapter than the preceding ones, and is tied together by the “reflection principle” (sometimes called the “mirror trick”), of reflecting part of a figure through a line to get a more tractable problem.

The book includes many problems, most asking for a proof, and about a quarter of the book is devoted to complete solutions to most of these problems, with hints for the rest.

Two somewhat similar books are Beckenbach & Bellman’s An Introduction to Inequalities (the preceding volume in the New Mathematical Library series), that studies many of the same problems but with algebraic rather than geometric methods; and Niven’s Maxima and Minima Without Calculus, that extends the ideas here to solve even more problems.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.