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

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:

• $S$ 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.

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

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.

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 $S$ may not exist if $S$ is not convex.
• Supporting line (supporting hyperplanes in $\mathbb {R} ^{2}$ )