In geometry, a supporting hyperplane of a set
S
in Euclidean space
R
n
is a hyperplane that has both of the following two properties:
S
is entirely contained in one of the two closed half-spaces bounded by the hyperplane
S
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
S
is a convex set in the topological vector space
X
=
R
n
,
and
x
0
is a point on the boundary of
S
,
then there exists a supporting hyperplane containing
x
0
.
If
x
∗
∈
X
∗
∖
{
0
}
(
X
∗
is the dual space of
X
,
x
∗
is a nonzero linear functional) such that
x
∗
(
x
0
)
≥
x
∗
(
x
)
for all
x
∈
S
, then
H
=
{
x
∈
X
:
x
∗
(
x
)
=
x
∗
(
x
0
)
}
defines a supporting hyperplane.
Conversely, if
S
is a closed set with nonempty interior such that every point on the boundary has a supporting hyperplane, then
S
is a convex set.
The hyperplane in the theorem may not be unique, as noticed in the second picture on the right. If the closed set
S
is not convex, the statement of the theorem is not true at all points on the boundary of
S
,
as illustrated in the third picture on the right.
The supporting hyperplanes of convex sets are also called tac-planes or tac-hyperplanes.
A related result is the separating hyperplane theorem, that every two disjoint convex sets can be separated by a hyperplane.
