Supporting hyperplane

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). 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: * 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. Here, a closed half-space is the half-space that includes the points within the hyperplane.

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). 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: 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. Here, a closed half-space is the half-space that includes the points within the hyperplane. ==Supporting hyperplane theorem== 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. 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, and is the intersection of all its supporting closed half-spaces. 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. The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes. The forward direction can be proved as a special case of the separating hyperplane theorem (see the page for the proof). For the converse direction, ==See also== Exposed face, the intersection of a set with a supporting hyperplane of it Support function Supporting line (supporting hyperplanes in \mathbb{R}^2 )

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