Supporting hyperplane

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

Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set $S$ in Euclidean space $\mathbb R^n$ if it meets both of the following:

• $S$ is entirely contained in one of the two closed half-spaces determined by the hyperplane
• $S$ has at least one point on the hyperplane.

Here, a closed half-space is the half-space that includes 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 closed convex set in a topological vector space $X,$ and $x_0$ is a point on the boundary of $S,$ then there exists a supporting hyperplane containing $x_0.$ If $x^* \in X^* \backslash \{0\}$ (the dual space of X) 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 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.

A related result is the separating hyperplane theorem.

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

• Ostaszewski, Adam (1990). Advanced mathematical methods. Cambridge; New York: Cambridge University Press. p. 129. ISBN 0521289645. 

• Giaquinta, Mariano; Hildebrandt, Stefan (1996). Calculus of variations. Berlin; New York: Springer. p. 57. ISBN 354050625X. 

• Goh, C. J.; Yang, X.Q. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. p. 13. ISBN 0415274796.