Supporting hyperplane

Summary

A supporting hyperplane of a set $S$ in $\R^n$ is a hyperplane that has all of $S$ on one side. It also touches $S$ at a point of the boundary of $S$. By the supporting hyperplane theorem, a convex set has a supporting hyperplane through each of its boundary points. Supporting hyperplanes link convex sets with linear functions: they describe where a linear function is largest on the set.