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 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 Euclidean space and x is a point on the boundary of S, then there exists a supporting hyperplane containing x.

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.

References

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

• 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.