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| | Greetings! I am Myrtle Shroyer. California is our birth place. Doing ceramics is what my family and I appreciate. In her professional life she is a payroll clerk but she's usually needed her personal business.<br><br>my site ... [http://www.youronlinepublishers.com/authWiki/AudreaocMalmrw www.youronlinepublishers.com] |
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| [[File:Supporting hyperplane1.svg|right|thumb|A [[convex set]] <math>S</math> (in pink), a supporting hyperplane of <math>S</math> (the dashed line), and the half-space delimited by the hyperplane which contains <math>S</math> (in light blue). ]]
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| '''Supporting hyperplane''' is a concept in [[geometry]]. A [[hyperplane]] divides a space into two [[Half-space (geometry)|half-space]]s. A hyperplane is said to '''support''' a [[Set (mathematics)|set]] <math>S</math> in [[Euclidean space]] <math>\mathbb R^n</math> if it meets both of the following:
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| * <math>S</math> is entirely contained in one of the two [[closed set|closed]] half-spaces determined by the hyperplane
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| * <math>S</math> has at least one boundary-point on the hyperplane.
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| Here, a closed half-space is the half-space that includes the hyperplane.
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| ==Supporting hyperplane theorem==
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| [[File:Supporting hyperplane2.svg|right|thumb|A convex set can have more than one supporting hyperplane at a given point on its boundary.]]
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| This [[theorem]] states that if <math>S</math> is a [[convex set]] in the [[topological vector space]] <math>X=\mathbb{R}^n,</math> and <math>x_0</math> is a point on the [[boundary (topology)|boundary]] of <math>S,</math> then there exists a supporting hyperplane containing <math>x_0.</math> If <math>x^* \in X^* \backslash \{0\}</math> (<math>X^*</math> is the [[dual space]] of <math>X</math>, <math>x^*</math> is a nonzero linear functional) such that <math>x^*\left(x_0\right) \geq x^*(x)</math> for all <math>x \in S</math>, then
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| :<math>H = \{x \in X: x^*(x) = x^*\left(x_0\right)\}</math>
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| defines a supporting hyperplane.<ref name="Boyd">{{cite book|title=Convex Optimization|first1=Stephen P.|last1=Boyd|first2=Lieven|last2=Vandenberghe|year=2004|publisher=Cambridge University Press|isbn=978-0-521-83378-3|url=http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf|format=pdf|accessdate=October 15, 2011|pages=50–51}}</ref>
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| Conversely, if <math>S</math> is a [[closed set]] with nonempty [[interior (topology)|interior]] such that every point on the boundary has a supporting hyperplane, then <math>S</math> is a convex set.<ref name="Boyd" />
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| The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set <math>S</math> is not convex, the statement of the theorem is not true at all points on the boundary of <math>S,</math> as illustrated in the third picture on the right.
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| A related result is the [[separating hyperplane theorem]].
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| ==See also==
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| [[File:Supporting hyperplane3.svg|right|thumb|A supporting hyperplane containing a given point on the boundary of <math>S</math> may not exist if <math>S</math> is not convex.]]
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| * [[Support function]]
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| ==References==
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| {{Reflist}}
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| *{{cite book
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| | last = Ostaszewski
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| | first = Adam
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| | title = Advanced mathematical methods
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| | publisher = Cambridge; New York: Cambridge University Press
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| | year = 1990
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| | isbn = 0-521-28964-5
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| | page = 129
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| }}
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| *{{cite book
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| | last = Giaquinta
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| | first = Mariano
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| | coauthors = Hildebrandt, Stefan
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| | title = Calculus of variations
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| | publisher = Berlin; New York: Springer
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| | year = 1996
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| | isbn = 3-540-50625-X
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| | page = 57
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| }}
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| *{{cite book
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| | last = Goh
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| | first = C. J.
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| | coauthors = Yang, X.Q.
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| | title = Duality in optimization and variational inequalities
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| | publisher = London; New York: Taylor & Francis
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| | year = 2002
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| | isbn = 0-415-27479-6
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| | page = 13
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| }}
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| [[Category:Convex geometry]]
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| [[Category:Functional analysis]]
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| [[Category:Duality theories]]
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| [[Category:Mathematical and quantitative methods (economics)]]
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Greetings! I am Myrtle Shroyer. California is our birth place. Doing ceramics is what my family and I appreciate. In her professional life she is a payroll clerk but she's usually needed her personal business.
my site ... www.youronlinepublishers.com