Theorem 18.8 in the book by Rockafellar establishes that any $n$-dimensional closed convex set $C$ in $R^n$ can be expressed as the intersection of the closed half spaces tangent to $C$. See here for the book page.
I'm having trouble seeing why the statement is true. For example, consider the square in $R^2$ which is the convex hull of $(1,1), (1,-1), (-1,1)$ and $(-1,-1)$. This is a closed convex set, but it does not seem to have any tangent hyperplanes (a hyperplane is tangent if it is the unique supporting hyperplane at a certain point), so how can the square be expressed as its tangent half spaces?
Is there something I'm missing here?
Edit: I realize that I might have some misunderstanding here, but I'm still not sure. On the same page of the book, tangent hyperplanes are mentioned as duals to the exposed points, that's why I thought for a hyperplane to be tangent it must only passes through one of the exposed points, which now I realize might not be the case. Does this mean that in the above example, hyperplanes containing one of the sides of the square are its tangent hyperplanes? This will rationalize the theorem for me, but can someone explain in what sense the tangent hyperplanes are dual to exposed points?
Thanks a lot!