Copolar convexity

TL;DR

This paper introduces copolar addition for convex sets in the positive orthant, proving covolume convexity and exploring implications for extremal functions using geodesic techniques in pluripotential theory.

Contribution

It presents a novel operation called copolar addition and demonstrates covolume convexity, linking convex geometry with pluripotential theory methods.

Findings

Covolumes of convex combinations are convex under copolar addition.

No relative extremal functions exist inside a non-constant geodesic between two toric extremal functions.

The proof employs geodesics of plurisubharmonic functions.

Abstract

We introduce a new operation, copolar addition, on unbounded convex subsets of the positive orthant of real euclidean space and establish convexity of the covolumes of the corresponding convex combinations. The proof is based on a technique of geodesics of plurisubharmonic functions. As an application, we show that there are no relative extremal functions inside a non-constant geodesic curve between two toric relative extremal functions.