Consider the following properties of a Banach space:
the intersection of any support hyperplane with the unit sphere is
(S) a singleton (this is the strict convexity);
(SF) finite-dimensional set;
(SC) compact in the norm topology.
It is easy to see that these properties are equivalent to the fact that any closed convex subset of a unit sphere is singleton/finite-dimensional/ compact.
Q1: Are (SF) and (SC) different?
Q2: Were these conditions considered in the literature? Do they have names?
Q3: Are there any necessary or sufficient conditions for them? In particular, are there any dual/predual conditions?