# Definitions Let $X$ be a vector space. A subset $K$ of $X$ is convex if whenever $x, y \in K$ and $0 \leq \lambda \leq 1$ we have $\lambda x+(1-\lambda) y \in K$. (Put more informally, a set is convex if the line segment joining any two points in the set is entirely contained in the set.) A function is said to be convex iff $\{ (x,y)\in \mathbb{R}^2: y>f(x) \}$ (the region strictly above its graph) is convex. # Properties See [[Convex twice differentiable real functions]].