Extreme contractions on finite-dimensional Banach spaces

We study extreme contractions in the setting of finite-dimensional polyhedral Banach spaces. Motivated by the famous Krein-Milman Theorem, we prove that a rank one norm one linear operator between such spaces can be expressed as a convex combination of rank one extreme contractions, whenever the domain is two-dimensional. We establish that the same result holds true in the space of all linear operators from ∞ⁿ(C) to ₁ⁿ (C). Furthermore, we present a geometric characterization of extreme contractions between finite-dimensional polyhedral Banach spaces.