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

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

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.^[1]

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.^[1]

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.^[2]

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

See also

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

• Support function

References

1. 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.
2. 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.

• 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.