Supporting hyperplane

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 hyperplaneHere, a closed half-space is the half-space that includes the hyperplane.

upporting hyperplane theorem

This theorem states that if $S$ is a closed convex set in Euclidean space $mathbb\; R^n,$ 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

*cite book
last = Ostaszewski
first = Adam
title = Advanced mathematical methods
publisher = Cambridge; New York: Cambridge University Press
date = 1990
pages = page 129
isbn = 0521289645

*cite book
last = Giaquinta
first = Mariano
coauthors = Hildebrandt, Stefan
title = Calculus of variations
publisher = Berlin; New York: Springer
date = 1996
pages = page 57
isbn = 354050625X

*cite book
last = Goh
first = C. J.
coauthors = Yang, X.Q.
title = Duality in optimization and variational inequalities
publisher = London; New York: Taylor & Francis
date = 2002
pages = page 13
isbn = 0415274796