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Supporting Hyperplane Theorem
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^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 \ (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 = \ defines a supporting hyperplane. Conversely, if S is a closed set In geometry, topology, and related branches of mathematics, a closed set is a Set (mathematics), set whose complement (set theory), complement is an open set. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Supporting Hyperplane1
Support may refer to: Arts, entertainment, and media * Supporting character * Support (art), a solid surface upon which a painting is executed Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a type of structural support to help prevent sideways movement * Structural support, architectural components that include arches, beams, columns, balconies, and stretchers Law and politics * Advocacy, in politics, support for constituencies, issues, or legislation * Lateral and subjacent support, a legal term Mathematics Mathematics (generally) * Support (mathematics), subset of the domain of a function where it is non-zero valued * Support (measure theory), a subset of a measurable space * Supporting hyperplane, sometimes referred to as support * Support of a module, a set of prime ideals in commutative algebra Statistics * Support, the natural logarithm of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
