Supporting hyperplane

In geometry, a supporting hyperplane of a set

in Euclidean space

is a hyperplane that has both of the following two properties:[1] 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

is a point on the boundary of

then there exists a supporting hyperplane containing

is the dual space of

is a nonzero linear functional) such that

, then defines a supporting hyperplane.

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.

[2] 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.

[3] 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, Define

to be the intersection of all its supporting closed half-spaces.

, show

{\displaystyle x\in \mathrm {int} (S)}

, and consider the line segment

be the largest number such that

{\displaystyle [x,t(y-x)+x]}

is contained in

{\displaystyle b=t(y-x)+x}

Draw a supporting hyperplane across

Let it be represented as a nonzero linear functional

{\displaystyle x\in \mathrm {int} (S)}