Plurisubharmonic geodesics and interpolating sets

TL;DR

This paper introduces a method for interpolating compact sets in complex space using geodesics of plurisubharmonic functions, revealing new geometric inequalities and descriptions of these interpolations.

Contribution

It develops a novel approach to interpolate compact subsets of complex space via plurisubharmonic geodesics and describes their properties using holomorphic hulls and capacity inequalities.

Findings

Interpolation sets are described as level sets of geodesic functions.

In the toric case, Monge-Ampère capacities satisfy a dual Brunn-Minkowski inequality.

Holomorphic hulls characterize the interpolating sets.

Abstract

We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of $C^n$. Namely, two non-pluripolar, polynomially closed, compact subsets of $C^n$ are interpolated as level sets $L_t=\{z: u_t(z)=-1\}$ for the geodesic $u_t$ between their relative extremal functions with respect to any ambient bounded domain. The sets $L_t$ are described in terms of certain holomorphic hulls. In the toric case, it is shown that the relative Monge-Amp\`ere capacities of $L_t$ satisfy a dual Brunn-Minkowski inequality.