I always had the impression that there was a duality (i.e. a contravariant equivalence of categories) between Banach spaces and certain notion of pointed compact convex set (something like algebras for a certain Giri monad acting on the category of compact topological spaces, maybe with additional conditions) given by:

To a Banach space one associate the unit ball of its dual in the weak* topology. And to a convex compact space one associate the space of linear function on it, with the uniform norm.

I can't remember if this is something I read somewhere or just something I imagined due to other weaker results in this direction (like the fact that every banach spaces is isomorphic to the space of Weak* continuous linear form on its dual).

So, do we indeed have a duality of this kind ? Has this been worked out somewhere, or is it some sort of folk results ? does it have a name ?

PS: I had too chose one answer to accept, but all three answers were all equally good and very interesting.