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?



A generalization of strict convexity
I think that one can answer Question 1 in the positive by studying the following construction: consider $X=\ell\_2\oplus \mathbb{R}$. Consider in $\ell\_2$ an infinitely dimensional compact ellipsoid $E$ centered at $0$ with all axes of length $<1$. Denote the unit ball of $\ell\_2$ by $B$. Consider in $X$ the norm whose unit ball is the closure of the convex hull of the union of three sets: $B\oplus \{0\}$, $E\oplus \{-1\}$, and $E\oplus \{1\}$. I would expect this to be (SC), but it is obviously not (SF). 


As for literature, you can check references listed in the book by Day, Normed Linear Spaces, book Deville-Godefroy-Zizler, and the survey of Godefroy in the "Handbook of the Geometry of Banach spaces". 


