- Supporting hyperplane
**Supporting hyperplane**is a concept ingeometry . Ahyperplane divides a space into twohalf-space s. A hyperplane is said to**support**a set $S$ inEuclidean 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 closedconvex set inEuclidean 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

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