Extremal structure in ultrapowers of Banach spaces

Given a bounded convex subset C of a Banach space X and a free ultrafilter u, we study which points (x(i))(u) are extreme points of the ultrapower C-u in X-u. In general, we obtain that when {x i } is made of extreme points (respectively denting points, strongly exposed points) and they satisfy some kind of uniformity, then (x(i))(u) is an extreme point (respectively denting point, strongly exposed point) of C-u. We also show that every extreme point of C-u is strongly extreme, and that every point exposed by a functional in (X*)(u) is strongly exposed, provided that u is a countably incomplete ultrafilter. Finally, we analyse the extremal structure of C-u in the case that C is a super weakly compact or uniformly convex set.