Interpolation of Weighted Extremal Functions

Abstract

An approach to interpolation of compact subsets of $${{\mathbb {C}}}^n$$ C n , including Brunn–Minkowski type inequalities for the capacities of the interpolating sets, was developed in [8] by means of plurisubharmonic geodesics between relative extremal functions of the given sets. Here we show that a much better control can be achieved by means of the geodesics between weighted relative extremal functions. In particular, we establish convexity properties of the capacities that are stronger than those given by the Brunn–Minkowski inequalities.

Plurisubharmonic Interpolation and Plurisubharmonic Geodesics

“…Theorem 10 ( [32]). Let K j D be polynomially convex, K 0 ∩ K 1 = ∅, and let the weights c j be chosen such that…”

“…For the local setting of plurisubharmonic functions on bounded domains, the geodesics were considered in the (unpublished) preprint [27] and, independently, in [28,29], and then in [4,[30][31][32][33]. Comparing with the compact manifold case, two main difficulties arise.…”