  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.
