Complex interpolation of $\mathbb{R}$-norms, duality and foliations

TL;DR

This paper extends complex interpolation methods to real Banach spaces and convex functions, revealing new duality properties via Legendre transform and linking to solutions of the complex Monge-Ampère equation.

Contribution

It introduces a novel interpolation framework for real Banach spaces and convex functions, generalizing classical complex methods and establishing new duality and geometric properties.

Findings

Generalized complex interpolation to real Banach spaces

Established duality via Legendre transform

Connected results to solutions of the complex Monge-Ampère equation

Abstract

The complex method of interpolation, going back to Calder\'on and Coifman et al., on the one hand, and the Alexander-Wermer-Slodkowski 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-Amp\`ere equation.