A note on the smoothness of the Minkowski function

Pranav Haridas; Jaikrishnan Janardhanan

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arXiv · 1805.11023

A note on the smoothness of the Minkowski function

Pranav Haridas·Jaikrishnan Janardhanan

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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.

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Pranav Haridas, Jaikrishnan Janardhanan. 2018-05-28. A note on the smoothness of the Minkowski function. https://arxiv.org/abs/1805.11023

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