View source for Supporting hyperplane

[[File:Supporting hyperplane1.svg|right|thumb|A [[Convex set|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|geometry]], a '''supporting hyperplane''' of a [[Set (mathematics)|set]] <math>S</math> in [[Euclidean space]] <math>\mathbb R^n</math> is a [[Hyperplane|hyperplane]] that has both of the following two properties:<ref>{{cite book |last=Luenberger |first=David G. |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 set|closed]] [[Half-space (geometry)|half-space]]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==
[[File:Supporting hyperplane2.svg|right|thumb|A convex set can have more than one supporting hyperplane at a given point on its boundary.]]
This [[Theorem|theorem]] states that if <math>S</math> is a [[Convex set|convex set]] in the [[Topological vector space|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|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
:<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|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, 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. (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 (see [[Hyperplane separation theorem#Case with possible intersections|the page for the proof]]). 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 S, 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}</math>, we have <math>f(y) < f(b)</math>, so <math>y \not\in T</math>. 
}}

==See also==
[[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.]]
* [[Support function]]
* [[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 theories]]

{{Sourceattribution|Supporting hyperplane}}