# Supporting hyperplane

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{{Short description|Hyperplane in geometry}}
[[File:Supporting hyperplane1.svg|right|thumb|A [convex set](/source/convex_set) <math>S</math> (in pink), a supporting hyperplane of <math>S</math> (the dashed line), and the supporting half-space delimited by the hyperplane which contains <math>S</math> (in light blue). ]]
In [geometry](/source/geometry), a '''supporting hyperplane''' of a [set](/source/Set_(mathematics)) <math>S</math> in [Euclidean space](/source/Euclidean_space) <math>\mathbb R^n</math> is a [hyperplane](/source/hyperplane) that has both of the following two properties:<ref>{{cite book |last=Luenberger |first=David G. |authorlink=David Luenberger |year=1969 |title=Optimization by Vector Space Methods |publisher=John Wiley & Sons |location=New York |isbn=978-0-471-18117-0 |url=https://books.google.com/books?id=lZU0CAH4RccC&pg=PA133 |page=133 }}</ref>
* <math>S</math> is entirely contained in one of the two [closed](/source/closed_set) [half-space](/source/Half-space_(geometry))s bounded by the hyperplane,
* <math>S</math> has at least one boundary-point on the hyperplane.
Here, a closed half-space is the half-space that includes the points within the hyperplane.

==Supporting hyperplane theorem==
right|thumb|A convex set can have more than one supporting hyperplane at a given point on its boundary.
This [theorem](/source/theorem) states that if <math>S</math> is a [convex set](/source/convex_set) in the [topological vector space](/source/topological_vector_space) <math>X=\mathbb{R}^n,</math> and <math>x_0</math> is a point on the [boundary](/source/boundary_(topology)) 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](/source/dual_space) of <math>X</math>, <math>x^*</math> is a nonzero [linear functional](/source/Linear_form)) such that <math>x^*\left(x_0\right) \geq x^*(x)</math> for all <math>x \in S</math>, then
:<math>H = \{x \in X: x^*(x) = x^*\left(x_0\right)\}</math>
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=https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf#page=64|format=pdf|accessdate=October 15, 2011|pages=50–51}}</ref>

Conversely, if <math>S</math> is a [closed set](/source/closed_set) with nonempty [interior](/source/interior_(topology)) such that every point on the boundary has a supporting hyperplane, then <math>S</math> is a convex set, and is the intersection of all its supporting closed half-spaces.<ref name="Boyd" />

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.

The supporting hyperplanes of convex sets are also called '''tac-planes''' or '''tac-hyperplanes'''.<ref>[Cassels, John W. S.](/source/John_W._S._Cassels) (1997), ''An Introduction to the Geometry of Numbers'', Springer Classics in Mathematics (reprint of 1959[3] and 1971 Springer-Verlag ed.), Springer-Verlag.</ref>

The forward direction can be proved as a special case of the [separating hyperplane theorem](/source/separating_hyperplane_theorem) (see [the page for the proof](/source/Hyperplane_separation_theorem)). For the converse direction,

{{Math proof|title=Proof|proof=

Define <math>T</math> to be the intersection of all its supporting closed half-spaces. Clearly <math>S \subset T</math>. Now let <math>y\not \in S</math>, show <math>y \not\in T</math>.  

Let <math>x\in \mathrm{int}(S)</math>, and consider the line segment <math>[x, y]</math>. Let <math>t</math> be the largest number such that <math>[x, t(y-x) + x]</math> is contained in <math>S</math>. Then <math>t\in (0, 1)</math>.  

Let <math>b = t(y-x) + x</math>, then <math>b\in \partial S</math>. Draw a supporting hyperplane across <math>b</math>. Let it be represented as a nonzero linear functional <math>f: \R^n \to \R</math> such that <math>\forall a\in T, f(a) \geq f(b)</math>. Then since <math>x\in \mathrm{int}(S)</math>, we have <math>f(x) > f(b)</math>. Thus by <math>\frac{f(y) - f(b)}{1-t} = \frac{f(b) - f(x)}{t-0} < 0</math>, we have <math>f(y) < f(b)</math>, so <math>y \not\in T</math>. 
}}

==See also==
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.
* [Support function](/source/Support_function)
* [Supporting line](/source/Supporting_line) (supporting hyperplanes in <math> \mathbb{R}^2</math>)

== Notes ==
{{Reflist}}
== References & further reading ==

*{{cite book
 | last       = Ostaszewski
 | first      = Adam
 | title      = Advanced mathematical methods
 | url       = https://archive.org/details/advancedmathemat0000osta
 | url-access = registration
 | publisher  = Cambridge; New York: Cambridge University Press
 | year       = 1990
 | isbn       = 0-521-28964-5
 | page      = [https://archive.org/details/advancedmathemat0000osta/page/129 129]
}}

*{{cite book
 | last       = Giaquinta
 | first      = Mariano
 |author2=Hildebrandt, Stefan 
  | title      = Calculus of variations
 | publisher  = Berlin; New York: Springer
 | year       = 1996
 | isbn       = 3-540-50625-X
 | page      = 57
}}

*{{cite book
 | last       = Goh
 | first      = C. J.
 |author2=Yang, X.Q.
  | title      = Duality in optimization and variational inequalities
 | publisher  = London; New York: Taylor & Francis
 | year       = 2002
 | isbn       = 0-415-27479-6
 | page      = 13
}}

*{{cite book
 | last       = Soltan
 | first      = V.
 | title      = Support and separation properties of convex sets in finite dimension
 | publisher  = Extracta Math. Vol. 36, no. 2, 241-278
 | year       = 2021
 }}

Category:Convex geometry
Category:Functional analysis
Category:Duality (mathematics)

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Adapted from the Wikipedia article [Supporting hyperplane](https://en.wikipedia.org/wiki/Supporting_hyperplane) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Supporting_hyperplane?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.