A note on the smoothness of the Minkowski function
Complex Variables
2018-05-29 v1
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.
Cite
@article{arxiv.1805.11023,
title = {A note on the smoothness of the Minkowski function},
author = {Pranav Haridas and Jaikrishnan Janardhanan},
journal= {arXiv preprint arXiv:1805.11023},
year = {2018}
}