- Supporting hyperplane
Supporting hyperplane is a concept in
geometry. A hyperplanedivides a space into two half-spaces. A hyperplane is said to support a set in Euclidean spaceif it meets both of the following:
* is entirely contained in one of the two closed half-spaces determined by the hyperplane
* has at least one point on the hyperplaneHere, a closed half-space is the half-space that includes the hyperplane.
upporting hyperplane theorem
theoremstates that if is a closed convex setin Euclidean spaceand is a point on the boundary of then there exists a supporting hyperplane containing
The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set is not convex, the statement of the theorem is not true at all points on the boundary of as illustrated in the third picture on the right.
A related result is the
separating hyperplane theorem.
last = Ostaszewski
first = Adam
title = Advanced mathematical methods
publisher = Cambridge; New York: Cambridge University Press
date = 1990
pages = page 129
isbn = 0521289645
last = Giaquinta
first = Mariano
coauthors = Hildebrandt, Stefan
title = Calculus of variations
publisher = Berlin; New York: Springer
date = 1996
pages = page 57
isbn = 354050625X
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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