Supporting hyperplane

Supporting hyperplane   (Redirected from Supporting hyperplane theorem) Jump to navigation Jump to search A convex set {displaystyle S} (in pink), a supporting hyperplane of {displaystyle S} (the dashed line), and the supporting half-space delimited by the hyperplane which contains {displaystyle S} (in light blue).

In geometry, a supporting hyperplane of a set {displaystyle S} in Euclidean space {displaystyle mathbb {R} ^{n}} is a hyperplane that has both of the following two properties:[1] {displaystyle S} is entirely contained in one of the two closed half-spaces bounded by the hyperplane, {displaystyle S} 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.

Contents 1 Supporting hyperplane theorem 2 See also 3 Notes 4 References & further reading Supporting hyperplane theorem A convex set can have more than one supporting hyperplane at a given point on its boundary.

This theorem states that if {displaystyle S} is a convex set in the topological vector space {displaystyle X=mathbb {R} ^{n},} and {displaystyle x_{0}} is a point on the boundary of {displaystyle S,} then there exists a supporting hyperplane containing {displaystyle x_{0}.} If {displaystyle x^{*}in X^{*}backslash {0}} ( {displaystyle X^{*}} is the dual space of {displaystyle X} , {displaystyle x^{*}} is a nonzero linear functional) such that {displaystyle x^{*}left(x_{0}right)geq x^{*}(x)} for all {displaystyle xin S} , then {displaystyle H={xin X:x^{*}(x)=x^{*}left(x_{0}right)}} defines a supporting hyperplane.[2] Conversely, if {displaystyle S} is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then {displaystyle S} is a convex set.[2] The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set {displaystyle S} is not convex, the statement of the theorem is not true at all points on the boundary of {displaystyle S,} as illustrated in the third picture on the right.

The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes.[3] A related result is the separating hyperplane theorem, that every two disjoint convex sets can be separated by a hyperplane.

See also A supporting hyperplane containing a given point on the boundary of {displaystyle S} may not exist if {displaystyle S} is not convex. Support function Supporting line (supporting hyperplanes in {displaystyle mathbb {R} ^{2}} ) Notes ^ Luenberger, David G. (1969). Optimization by Vector Space Methods. New York: John Wiley & Sons. p. 133. ISBN 978-0-471-18117-0. ^ Jump up to: a b Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization (pdf). Cambridge University Press. pp. 50–51. ISBN 978-0-521-83378-3. Retrieved October 15, 2011. ^ 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. References & further reading Ostaszewski, Adam (1990). Advanced mathematical methods. Cambridge; New York: Cambridge University Press. p. 129. ISBN 0-521-28964-5. Giaquinta, Mariano; Hildebrandt, Stefan (1996). Calculus of variations. Berlin; New York: Springer. p. 57. ISBN 3-540-50625-X. Goh, C. J.; Yang, X.Q. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. p. 13. ISBN 0-415-27479-6. Categories: Convex geometryFunctional analysisDuality theories