A note on the smoothness of the Minkowski function

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.

Keywords:

• Minkowski function
• balanced domains
• quasi-balanced domains
• plurisubharmonic defining function

AMS SUBJECT CLASSIFICATION:

• Primary: 32A07

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Pranav Haridas is supported by a grant provided by the National Board for Higher Mathematics, India. Jaikrishnan Janardhanan is supported by a DST-INSPIRE fellowship from the Department of Science and Technology, India.