Consider the following property of a Banach space $E$: intersection of any support hyperplane with the unit sphere is compact in the norm topology (recall that if we replace "compact" with "singleton" we get strict convexity).

IT is easy to see that this is equivalent to the fact that any convex subsets of a unit sphere is compact.

Does this condition have a name? Is there any necessary or sufficient condition for it? In particular, is there any dual/predual condition?

Is it actually possible that these intersections are not finite-dimensional convex sets?