In geometry, a supporting hyperplane of a set ${\displaystyle S}$ in Euclidean space ${\displaystyle \mathbb {R} ^{n))$ is a hyperplane that has both of the following two properties:^[1]

• ${\displaystyle S}$ is entirely contained in one of the two closed half-spaces bounded by the hyperplane,
• ${\displaystyle S}$ has at least one boundary-point on the hyperplane.

Here, a closed half-space is the half-space that includes the points within the hyperplane.

Supporting hyperplane theorem

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

${\displaystyle H=\{x\in X:x^{*}(x)=x^{*}\left(x_{0}\right)\))$

defines a supporting hyperplane.^[2]

Conversely, if ${\displaystyle S}$ is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then ${\displaystyle S}$ is a convex set, and is the intersection of all its supporting closed half-spaces.^[2]

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

The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes.^[3]

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,