We investigate the concepts of linear convexity and ℂ-convexity in complex Banach spaces. The main result is that any ℂ-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a ℂ-convex domain Ω in the Banach space X and a point p∉Ω, there is a complex hyperplane through p that does not intersect Ω. We also prove that linearly convex domains are holomorphically convex, and that Kergin interpolation can be performed on holomorphic mappings defined in ℂ-convex domains.