2. Convexity

This chapter starts with the study of basic properties of convex sets, including the separation and the supporting hyperplane theorems. Next, we introduce a class of functions that are very useful in optimization, the convex functions, for which the local minima are also global minima. We begin with functions of one variable and follow with functions of several variables (or multivariate). The chapter finishes with the study of two particular classes of convex functions, the strictly convex and the strongly convex functions, which are introduced in order to ensure the existence and uniqueness of global minimum. Taking into account the instrumental character of this chapter, we have preferred to sacrifice generality to simplicity in the presentation of some results; compare, e.g., Propositions 2.33, 2.37, and 2.60 below with [80, Theorem 23.4], [80, Theorem 24.7], and [25, Theorem 2.8], respectively.