Supporting hyperplane

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Supporting hyperplane of a set, S. A hyperplane that contains S in one of its closed halfspaces and intersects the closure of S with at least one point.

            ______________________  supporting hyperplane (line)                     /\                    /  \                   / S  \                  /______\

Suppose S is closed and convex. A key fact is that every supporting hyperplane contains an extreme point of S. If S is a polyhedron, the facets define a finite collection of supporting hyperplanes that completely determine the polyhedron (as the intersection of the associated halfspaces that contain S).

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