Sharp corner points and isometric extension problem in Banach spaces

In this article, we begin using some geometric methods to study the isometric extension problem in general real Banach spaces. For any Banach space Y, we define a collection of “sharp corner points” of the unit ball B1(Y∗), which is empty if Y is strictly convex and dimY≥2. Then we prove that any surjective isometry between two unit spheres of Banach spaces X and Y has a linear isometric extension on the whole space if Y is a Gâteaux differentiability space (in particular, separable spaces or reflexive spaces) and the intersection of “sharp corner points” and weak∗-exposed points of B(Y∗) is weak∗-dense in the latter. Moreover, we study the “sharp corner points” in many classical Banach spaces and solve isometric extension problem affirmatively in the case that Y is (ℓ1),c0(Γ),c(Γ),ℓ∞(Γ) or some C(Ω).