Convex function on R^n

Definition

A set $C \subseteq \mathbb{R}^n$ is convex if $t x + (1 - t) y \in C$ for all $x, y \in C$ and all $t \in [0, 1]$. Let $C \subseteq \mathbb{R}^n$ be convex. A function $f \colon C \to \mathbb{R}$ is convex if, for all $x, y \in C$ and all $t \in [0, 1]$, $$f(t x + (1 - t) y) \le t f(x) + (1 - t) f(y).$$ The function $f$ is strictly convex if the inequality is strict for all $x \ne y$ and all $t \in (0, 1)$.

Remarks

If $C$ is open and $f$ is differentiable on $C$, then $f$ is convex if and only if $f(y) \ge f(x) + \nabla f(x) \cdot (y - x)$ for all $x, y \in C$. Let $C$ be open and let $f$ be of class $C^2$. Then $f$ is convex if and only if its Hessian matrix is positive semidefinite at each point of $C$. Each local minimum of a convex function is a global minimum, and a strictly convex function has at most one global minimum. A function can be convex in each variable separately but not convex. The function $f(x, y) = x y$ is linear in each variable, but $f(t, -t) = -t^2$ is not convex.

Explanations

L1 human

A convex function is bowl-shaped: the straight segment between two points of its graph never goes below the graph. Examples are $\|x\|^2$ and, on all of $\mathbb{R}^n$, each linear function.