In convex analysis and mathematical optimization, the supporting functional is a generalization of the supporting hyperplane of a set.
Let X be a locally convex topological space, and
C
⊂
X
be a convex set, then the continuous linear functional
ϕ
:
X
→
R
is a supporting functional of C at the point
x
0
if
ϕ
(
x
)
≤
ϕ
(
x
0
)
for every
x
∈
C
.
If
h
C
:
X
∗
→
R
(where
X
∗
is the dual space of
X
) is a support function of the set C, then if
h
C
(
x
∗
)
=
x
∗
(
x
0
)
, it follows that
h
C
defines a supporting functional
ϕ
:
X
→
R
of C at the point
x
0
such that
ϕ
(
x
)
=
x
∗
(
x
)
for any
x
∈
X
.
If
ϕ
is a supporting functional of the convex set C at the point
x
0
∈
C
such that
ϕ
(
x
0
)
=
σ
=
sup
x
∈
C
ϕ
(
x
)
>
inf
x
∈
C
ϕ
(
x
)
then
H
=
ϕ
−
1
(
σ
)
defines a supporting hyperplane to C at
x
0
.
