In geometry, a supporting hyperplane of a set S in Euclidean space \( \mathbb {R} ^{n} \) is a hyperplane that has both of the following two properties:[1]
S is entirely contained in one of the two closed half-spaces bounded by the hyperplane,
S has at least one boundary-point on the hyperplane.
A convex set S (in pink), a supporting hyperplane of S (the dashed line), and the supporting half-space delimited by the hyperplane which contains S (in light blue).
Here, a closed half-space is the half-space that includes the points within the hyperplane.
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 S is a convex set in the topological vector space \( X={\mathbb {R}}^{n} \), and \( x_{0} \) is a point on the boundary of S , then there exists a supporting hyperplane containing \( x_{0} \). If \( x^{*}\in X^{*}\backslash \{0\} \) ( \( X^{*} \) is the dual space of X, \( x^{*} \) is a nonzero linear functional) such that \( x^{*}\left(x_{0}\right)\geq x^{*}(x) \) for all \( x\in S \), then
\( H=\{x\in X:x^{*}(x)=x^{*}\left(x_{0}\right)\} \)
defines a supporting hyperplane.[2]
Conversely, if S is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then 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 S is not convex, the statement of the theorem is not true at all points on the boundary of S 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 S may not exist if 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.
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.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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