Supporting hyperplane

Supporting hyperplane is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to support a set $S$ in Euclidean space $\mathbb {R} ^{n}$ if it meets both of the following:

$S$ is entirely contained in one of the two closed half-spaces determined by the hyperplane
$S$ has at least one point on the hyperplane.

Here, a closed half-space is the half-space that includes the hyperplane.

Supporting hyperplane theorem

This theorem states that if $S$ is a convex set in a topological vector space $X,$ 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.^[1]

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.^[1]

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 $S,$ as illustrated in the third picture on the right.

A related result is the separating hyperplane theorem.

References

^ ^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.

Ostaszewski, Adam (1990). Advanced mathematical methods. Cambridge; New York: Cambridge University Press. p. 129. ISBN 0-521-28964-5.

Giaquinta, Mariano (1996). Calculus of variations. Berlin; New York: Springer. p. 57. ISBN 3-540-50625-X. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

Goh, C. J. (2002). Duality in optimization and variational inequalities. London; New York: Taylor & Francis. p. 13. ISBN 0-415-27479-6. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

[Boyd-1] 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.

[1]

Supporting hyperplane theorem

See also

References