The following statement is a direct consequence of the definitions.

Proposition
(Polar cone properties). For any nonempty
set
,
we have

1.
is a closed convex set.

2.
.

3. If
for some set
then
.

Proof
First, we show that for any nonempty
we have
.
Indeed, by the definitions, for a fixed
Therefore,
.

Next, we prove that for a closed nonempty
,
we have
.

Let
.
Since
is closed, there exists the projection
.
Let us translate the coordinate system so that
.
Then by the proposition (
Projection
theorem
)-2 we
have
Hence,
We already established that
Therefore,
but
also
Hence, for a nonempty set
(empty
is a trivial case) we
have
By the definition of polar cone, we always have
for a nonempty
.
Hence,
.

Finally, we prove that
.
By the proposition (
Polar cone
properties
), we have
Therefore,
We already proved that
.