Niels Lauritzen’s new book Undergraduate Convexity has an intriguing title. What about such a book would make it suitable for undergraduates? Convexity is not often taught as its own subject, especially at an undergraduate level. But convexity is an important concept — it is a common form of nonlinearity, and often the most tractable form — and perhaps should be given its own course more often.
To assume that undergraduate in the title means elementary would be somewhat misleading. The book is not so much elementary as concrete. It is filled with computational examples and exercises, mostly computations that can be carried out by hand. A graduate student wanting more hands-on experience with convexity could use this book to complement a more theoretical book long on theorems and proofs but short on algorithms and examples. And while Undergraduate Convexity stresses calculations, it does not skimp on theory. It has a number of theorems not always included in books on convexity, particularly results on polyhedra.
The book is divided evenly into two parts, devoted to convex sets and convex functions. The former is more substantial because the latter devotes a fair amount of space to reviewing multivariate calculus. This makes the book more self-contained, but it uses up space that could have been used to discuss convex functions in more depth. Also, much of the section on convex functions is more specifically about the optimization of convex functions.
Undergraduate Convexity would make an excellent textbook. An instructor might choose to have students present some of the examples while he or she provides commentary, perhaps alternating coaching and lecturing. A course taught from this book could be a good transition into more abstract mathematics, exposing students to general theory then giving them the familiar comfort of more computational exercises. One could also use the book as a warm-up to a more advanced course in optimization.
John D. Cook is an independent consultant and blogs at The Endeavour.