Sorry, I really mean $\pi\pi^*\pi$, as in the original posting. Sorry about the confusion...

I really mean $\pi^*\pi\pi^*$. And $\pi^*$ is what I defined, a fixed permutation independent of $\pi$. Thanks for the hint about involutions!

One can bypass the ``lsc implies Baire-1'' step by noting directly that Angelo's $f$ is Baire-1, since one has for $g_n = \sup\{f_1,\dots,f_n\}$ that $f=\lim g_n$ pointwise.

The theorem says that C has the RNP iff every closed convex bounded subset is the closed convex hull of its denting points iff every closed convex bounded subset is the closed convex hull of its strongly exposed points. This leaves the possibility open that C is the closed convex hull of its denting points without having a single strongly exposed point. (Of course, such a C won't have the RNP). Unfortunately, I don't have an example to this effect at hand...

@Uri The complex case is much harder, see E. Behrends's paper in Math. Ann. 290, No. 3, 463-471 (1991).