Complex interpolation of R-norms, duality and foliations

Journal article, 2020

The complex method of interpolation, going back to Calderón and Coifman et al., on the one hand, and the Alexander–Wermer–Słodkowski theorem on polynomial hulls with convex fibers, on the other hand, are generalized to a method of interpolation of real (finite-dimensional) Banach spaces and of convex functions. The underlying duality in this method is given by the Legendre transform. Our results can also be interpreted as new properties of solutions of the homogeneous complex Monge–Ampère equation.