# Supporting hyperplane

A convex set ${\displaystyle S}$ (in pink), a supporting hyperplane of ${\displaystyle S}$ (the dashed line), and the supporting half-space delimited by the hyperplane which contains ${\displaystyle S}$ (in light blue).

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:

• ${\displaystyle S}$ 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

A convex set can have more than one supporting hyperplane at a given point on its boundary.

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.[1]

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.[1]

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.[2]

A related result is the separating hyperplane theorem, that every two disjoint convex sets can be separated by a hyperplane.

A supporting hyperplane containing a given point on the boundary of ${\displaystyle S}$ may not exist if ${\displaystyle S}$ is not convex.