# Interpolation of weighted extremal functions

## TL;DR

This paper improves the understanding of complex interpolation of compact sets in complex space by using geodesics between weighted extremal functions, leading to stronger convexity properties of capacities than traditional inequalities.

## Contribution

It introduces a refined method using weighted relative extremal functions for better control and stronger convexity results in the interpolation of compact sets.

## Key findings

- Establishment of convexity properties of capacities beyond Brunn-Minkowski inequalities
- Demonstration of improved control via geodesics between weighted extremal functions
- Enhanced inequalities for capacities of interpolating sets

## Abstract

An approach to complex interpolation of compact subsets of $\Bbb C^n$, including Brunn-Minkowski type inequalities for the capacities of the interpolating sets, was developed recently 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.