In mathematics, the bipolar theorem is a theorem in convex analysis which provides necessary and sufficient conditions for a cone to be equal to its bipolar. The bipolar theorem can be seen as a special case of the Fenchel–Moreau theorem.
For any nonempty set
C
⊂
X
in some linear space
X
, then the bipolar cone
C
o
o
=
(
C
o
)
o
is given by
C
o
o
=
cl
(
co
{
λ
c
:
λ
≥
0
,
c
∈
C
}
)
where
co
denotes the convex hull.
C
⊂
X
is a nonempty closed convex cone if and only if
C
+
+
=
C
o
o
=
C
when
C
+
+
=
(
C
+
)
+
, where
(
⋅
)
+
denotes the positive dual cone.
Or more generally, if
C
is a nonempty convex cone then the bipolar cone is given by
C
o
o
=
cl
C
.
If
f
(
x
)
=
δ
(
x
|
C
)
=
{
0
if
x
∈
C
+
∞
else
is the indicator function for a cone
C
. Then the convex conjugate
f
∗
(
x
∗
)
=
δ
(
x
∗
|
C
o
)
=
δ
∗
(
x
∗
|
C
)
=
sup
x
∈
C
⟨
x
∗
,
x
⟩
is the support function for
C
, and
f
∗
∗
(
x
)
=
δ
(
x
|
C
o
o
)
. Therefore,
C
=
C
o
o
if and only if
f
=
f
∗
∗
.
