Supporting hyperplane

In geometry, a supporting hyperplane of a set in Euclidean space is a hyperplane that has both of the following two properties: is entirely contained in one of the two closed half-spaces bounded by the hyperplane, 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. This theorem states that if is a convex set in the topological vector space and is a point on the boundary of then there exists a supporting hyperplane containing If ( is the dual space of , is a nonzero linear functional) such that for all , then defines a supporting hyperplane. Conversely, if is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then 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 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. 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).