Supporting hyperplane

In geometry, a supporting hyperplane of a set $$S$$ in Euclidean space $$\mathbb R^n$$ is a hyperplane that has both of the following two properties: Here, a closed half-space is the half-space that includes the points within the hyperplane.
 * $$S$$ is entirely contained in one of the two closed half-spaces bounded by the hyperplane,
 * $$S$$ has at least one boundary-point on the hyperplane.

Supporting hyperplane theorem
This theorem states that if $$S$$ is a convex set in the topological vector space $$X=\mathbb{R}^n,$$ and $$x_0$$ is a point on the boundary of $$S,$$ then there exists a supporting hyperplane containing $$x_0.$$  If $$x^* \in X^* \backslash \{0\}$$ ($$X^*$$ is the dual space of $$X$$, $$x^*$$ is a nonzero linear functional) such that $$x^*\left(x_0\right) \geq x^*(x)$$ for all $$x \in S$$, then
 * $$H = \{x \in X: x^*(x) = x^*\left(x_0\right)\}$$

defines a supporting hyperplane.

Conversely, if $$S$$ is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then $$S$$ is a convex set, and is the intersection of all its supporting closed half-spaces.

The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set $$S$$ is not convex, the statement of the theorem is not true at all points on the boundary of $$S,$$ as illustrated in the third picture on the right.

The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes.

The forward direction can be proved as a special case of the separating hyperplane theorem (see the page for the proof). For the converse direction,

$$

References & further reading