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EspañolSUPPORTING HYPERPLANE
(Redirected from Supporting hyperplane theorem)
'Supporting hyperplane' is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to 'support' a set in Euclidean space if it meets both of the following:
★ is entirely contained in one of the two closed half-spaces of the hyperplane
★ has at least one point on the hyperplane
Here, a closed half-space is the half-space that includes the hyperplane.
This theorem states that if is a closed convex set in Euclidean space and is a point on the boundary of then there exists a supporting hyperplane containing
The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set is not convex, the statement of the theorem is not true at all points on the boundary of as illustrated in the third picture on the right.
A related result is the separating hyperplane theorem.
★ Advanced mathematical methods, , Adam, Ostaszewski, Cambridge; New York: Cambridge University Press, ,
★ Calculus of variations, , Mariano, Giaquinta, Berlin; New York: Springer, ,
★ Duality in optimization and variational inequalities, , C. J., Goh, London; New York: Taylor & Francis, ,
A convex set (in pink), a supporting hyperplane of (the dashed line), and the half-space delimited by the hyperplane which contains (in light blue).
'Supporting hyperplane' is a concept in geometry. A hyperplane divides a space into two half-spaces. A hyperplane is said to 'support' a set in Euclidean space if it meets both of the following:
★ is entirely contained in one of the two closed half-spaces of the hyperplane
★ has at least one point on the hyperplane
Here, a closed half-space is the half-space that includes the hyperplane.
| Contents |
| Supporting hyperplane theorem |
| References |
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 is a closed convex set in Euclidean space and is a point on the boundary of then there exists a supporting hyperplane containing
The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set is not convex, the statement of the theorem is not true at all points on the boundary of as illustrated in the third picture on the right.
A related result is the separating hyperplane theorem.
References
A supporting hyperplane containing a given point on the boundary of may not exist if is not convex.
★ Advanced mathematical methods, , Adam, Ostaszewski, Cambridge; New York: Cambridge University Press, ,
★ Calculus of variations, , Mariano, Giaquinta, Berlin; New York: Springer, ,
★ Duality in optimization and variational inequalities, , C. J., Goh, London; New York: Taylor & Francis, ,
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