Supporting hyperplane of a convex function

A supporting hyperplane of a convex function $f$ at a point $x\in\operatorname{dom} f$ (see domain) is an affine function

\ell(y)=f(x)+\langle g,\,y-x\rangle

such that $\ell(y)\le f(y)$ for all $y\in\mathbb{R}^n$ .

Equivalently, the graph $\{(y,t): t=\ell(y)\}$ is a hyperplane in $\mathbb{R}^{n+1}$ that supports the epigraph of $f$ (a convex set when $f$ is convex). Such supporting hyperplanes are in one-to-one correspondence with subgradients: $g$ supports $f$ at $x$ exactly when $g\in\partial f(x)$ (see subdifferential).

Examples

For $f(x)=x^2$ on $\mathbb{R}$ and any $x_0$ , the tangent line $\ell(x)=x_0^2+2x_0(x-x_0)$ is a supporting hyperplane at $x_0$ .
For $f(x)=|x|$ on $\mathbb{R}$ at $x=0$ , every line $\ell(x)=gx$ with $g\in[-1,1]$ is a supporting hyperplane.