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.