Extremal structure in ultrapowers of Banach spaces

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