# The polar of a set: Importance and Applications

Given a duality $(X,Y)$, for any subset $A\subset X$, we can define the polar set $A^{\circ}\subset Y$. In this sense, the polar relates sets and sets in the dual space. Since cones are sets, we can also do this for cones and get the polar cone. Here, we can also define the dual cone, which is somehow the "natural" dual cone in the dual of the underlying linear space.

I have wondered for a time: What use does the polar set serve? Whenever I saw this defined in a book or a lecture, it seemed to be a very important concept, however, I have never actually seen it been used very much. It is only ever covered as a "side topic".

Of course, there is the bipolar theorem and I have heard this theorem can be used to give a proof of the Banach-Alaoglu theorem, but what else?

Thus my question: Why is the polar set useful? Why is it conceptually important?

It is very important in optimization. In fact, if you have to minimize $f(x)$ over a convex, closed set $C \subset X$, the first order optimality condition reads $$f'(x) \in T_C(x)^\circ.$$ Here $T_C(x)$ is the tangent cone and its polar is also known as the normal cone.