Chapter 3 - Convex Functions

Abstract

Convexity plays a central role in economic models. It is a strong property of a real-valued function because it implies continuity and existence of one-sided gradients on the interior of the domain, and existence of gradients almost everywhere. A function is convex on an open set if and only if the tangents of the function lie everywhere below the function. When a function is also homogeneous, as in some economic models, the tangents pass through the origin. When a function has a Hessian everywhere, the function is convex if and only if the Hessian is positive semidefinite everywhere. Some functions that arise in economic models have a weaker property called quasiconvexity. A function is quasiconvex if and only if its lower contour sets are convex.