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-spacess 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 is a convex set in the topological vector space is a point on the boundary is the dual space is a nonzero linear functional, then

defines a supporting hyperplane.

is a closed set with nonempty interior is a convex set, and is the intersection of all its supporting closed half-spaces.

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)

Notes

References & further reading