{{Short description|Hyperplane in geometry}} [[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 supporting half-space delimited by the hyperplane which contains <math>S</math> (in light blue). ]] In geometry, a '''supporting hyperplane''' of a set <math>S</math> in Euclidean space <math>\mathbb R^n</math> is a 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 half-spaces 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 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 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 :<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 with nonempty 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 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 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 * 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)