113 Geometric Inequalities from the AwesomeMath Summer Program

Within the world of inequalities, geometric inequalities are usually not treated as a separate subject. Most or all of the books that focus on geometric inequalities, including this one, are aimed at high school students (a recent example is Leng’s Geometric Inequalities and and oldie but goodie is Kazarinoff’s Geometric Inequalities).

The present book deals with a wide variety of problems stated in geometric terms. There is also a variety of solution methods, but most problems are not solved by geometry — they are solved by converting the problem into an algebraic form (often by trigonometry) and then solving the problem by applying familiar algebraic inequalities such as the Arithmetic Mean-Geometric Mean inequality or the Cauchy-Schwarz inequality.

The book begins with three expository chapters. The first focuses on “elementary” problems, which here means those that can be solved by the triangle inequality or by expressing the quantities using trigonometry and observing that the sine and cosine are always less than or equal to 1 in absolute value. The second chapter is not geometric at all, but is an excursion into algebraic inequalities that will be used later. The third chapter concentrates on problems involving triangles. The exposition in these chapters is primarily a series of worked problems, with a little bit of connecting narrative. Then there are two chapters (elementary and advanced) of 113 problems to solve, followed by worked solutions.

Like other books in the AwesomeMath series, this one focuses on problems of the sort that appear in the International Mathematical Olympiad, and the book seems aimed at students preparing for this competition. I think this is a reasonable book for this purpose. Its greatest weakness is the narrative; it reads like a disconnected series of problems and it’s hard to see the overall outlines of the subject. One conspicuous absence here is isoperimetric problems, which seem not to be mentioned at all. For people specifically interested in geometric inequalities, Kazarinoff’s book is a better choice. It has better explanations and a more purely geometrical approach. Another interesting book, that overlaps wth these two but focuses on extremal problems rather than inequalities, and is very geometrical, is Niven’s Maxima and Minima Without Calculus.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.