We study AB(n) miktoarm star block copolymers in the strong segregation limit, focusing on the role that the AB interface plays in determining the phase behavior. We develop an extension of the kinked-path approach which allows us to explore the energetic dependence on interfacial shape. We consider a one-parameter family of interfaces to study the columnar to lamellar transition in asymmetric stars. We compare with recent experimental results. We discuss the stability of the A15 lattice of spherelike micelles in the context of interfacial energy minimization. We corroborate our theory by implementing a numerically exact self-consistent-field theory to probe the phase diagram and the shape of the AB interface.