A note on the smoothness of the Minkowski function

TL;DR

This paper proves that the Minkowski function of a smooth, bounded, pseudoconvex quasi-balanced domain is smooth away from the origin, enabling the construction of smooth plurisubharmonic defining functions.

Contribution

It establishes the smoothness of the Minkowski function under boundary smoothness assumptions, providing a new tool for analyzing quasi-balanced domains.

Findings

Minkowski function is smooth away from the origin for smooth, bounded, pseudoconvex quasi-balanced domains.

Allows construction of smooth plurisubharmonic defining functions for such domains.

Result is new even for balanced domains.

Abstract

The Minkowski function is a crucial tool used in the study of balanced domains and, more generally, quasi-balanced domains in several complex variables. If a quasi-balanced domain is bounded and pseudoconvex then it is well-known that its Minkowski function is plurisubharmonic. In this short note, we prove that under the additional assumption of smoothness of the boundary, the Minkowski function of a quasi-balanced domain is in fact smooth away from the origin. This allows us to construct a smooth plurisubharmonic defining function for such domains. Our result is new even in the case of balanced domains.