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Supporting hyperplane

Hyperplane in geometry

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Source updated June 29, 2026

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.

01Supporting 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,

Proof

Define $T$ to be the intersection of all its supporting closed half-spaces. Clearly $S\subset T$ . Now let $y\not \in S$ , show $y\not \in T$ .

Let $x\in \mathrm {int} (S)$ , and consider the line segment $[x,y]$ . Let $t$ be the largest number such that $[x,t(y-x)+x]$ is contained in $S$ . Then $t\in (0,1)$ .

Let $b=t(y-x)+x$ , then $b\in \partial S$ . Draw a supporting hyperplane across $b$ . Let it be represented as a nonzero linear functional $f:\mathbb {R} ^{n}\to \mathbb {R}$ such that $\forall a\in T,f(a)\geq f(b)$ . Then since $x\in \mathrm {int} (S)$ , we have $f(x)>f(b)$ . Thus by ${\frac {f(y)-f(b)}{1-t}}={\frac {f(b)-f(x)}{t-0}}<0$ , we have $f(y)<f(b)$ , so $y\not \in T$ .

02References & further reading

Ostaszewski, Adam (1990). Advanced mathematical methods, London School of Economics Mathematics Series. Cambridge; New York: Cambridge University Press. p. 129. ISBN 0-521-28964-5.

Giaquinta, Mariano; Hildebrandt, Stefan (1996). Calculus of variations. Berlin; New York: Springer. p. 57. ISBN 3-540-50625-X.

Goh, C. J.; Yang, X.Q. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. p. 13. ISBN 0-415-27479-6.

Soltan, V. (2021). Support and separation properties of convex sets in finite dimension. Extracta Math. Vol. 36, no. 2, 241-278.

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Sources and credits

This article is adapted from the Wikipedia article “Supporting hyperplane”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Supporting hyperplane1.svg by Oleg Alexandrov, Public domain
Supporting hyperplane2.svg by Oleg Alexandrov, Public domain
Supporting hyperplane3.svg by Oleg Alexandrov, Public domain

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Supporting hyperplane

01Supporting hyperplane theorem

02References & further reading

Sources and credits

Related topics

Support function

Supporting line