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:

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

Supporting hyperplane theorem

This theorem states that if S is a closed 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\} (the dual space of X) 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 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.

See also

References

  1. ^ a b Boyd, Stephen P.; Vandenberghe, Lieven (2004) (pdf). Convex Optimization. Cambridge University Press. pp. 50–51. ISBN 9780521833783. http://www.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf. Retrieved October 15, 2011.