Convex Analysis

Basics

A convex set is a set \(X\) such that \(\forall x_1, x_2 \in X, \forall \lambda \in [0, 1], \lambda x_1 + (1 - \lambda) x_2 \in X\).

A convex function is a real-valued function \(f: X \rightarrow \mathbb{R}\) such that \(\forall x_1, x_2 \in X, \forall \lambda \in [0, 1], f(\lambda x_1 + (1 - \lambda) x_2) \leq \lambda f(x_1) + (1 - \lambda) f(x_2)\) i.e. Jensen’s Inequality. Interestingly, Wikipedia says that another definition of a convex function is a function that satisfies Jensen’s inequality.