title: Supporting hyperplane
text: 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.
  
  Supporting hyperplane theorem
  
  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). For the converse direction,
  
  See also
  
   Support function
   Supporting line (supporting hyperplanes in )
  
  Notes
  
  References & further reading 
  
  Convex geometry
  Functional analysis
  Duality theories