Supporting hyperplane

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.

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).

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.^[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,

Proof

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

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

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

See also

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

• Support function
• Supporting line (supporting hyperplanes in ${\displaystyle \mathbb {R} ^{2}}$)

Notes

1. Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. p. 133. ISBN 978-0-471-18117-0.
2. Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 50–51. ISBN 978-0-521-83378-3. Retrieved October 15, 2011.
3. Cassels, John W. S. (1997), An Introduction to the Geometry of Numbers, Springer Classics in Mathematics (reprint of 1959[3] and 1971 Springer-Verlag ed.), Springer-Verlag.

References &amp; further reading

• Ostaszewski, Adam (1990). Advanced mathematical methods. 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.