When my father was alive, I often heard the words, “Niki, I have problem”; and more often than not the question which unfolded on our living-room blackboard dealt with an inequality. Nowadays, I like to think that it was partly because I never encountered questions at school which were even remotely similar to those with which I wrestled at home that I almost never found the solutions to any of the problems on the home blackboard. Mathematical curricula of today's secondary schools continue to ignore the topic of inequalities. Yet every mathematician knows that inequalities are important in all branches of mathematics, sometimes even more important than equalities.

In 1958 the Ann Arbor Public Schools gave me the opportunity to hold frequent mathematical discussions with an enthusiastic group of young people. These students, by their response and interest, stimulated me to write the present book. Their understanding and enjoyment of inequalities led me to believe that a careful exposition of some of the topics we discussed would be well received by a wider audience.

Geometric inequalities are especially appealing because their statements can be easily grasped; at the same time they provide an excellent introduction to creative mathematical thought and to the spirit of modern mathematics. The elementary inequalities that form the subject matter of this book have the further advantage of demanding and requiring only a clear head and a minimum of formal mathematical training in order to be understood: a year of high-school algebra and the fundamentals of plane geometry will usually be sufficient.