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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关于 Minkowski 函数平滑度的注记

Minkowski 函数是用于研究平衡域的重要工具，更一般地说，是研究多个复杂变量中的准平衡域。如果一个准平衡域是有界和伪凸的，那么众所周知，它的闵可夫斯基函数是多次谐波的。在这个简短的说明中，我们证明了在边界平滑的附加假设下，准平衡域的 Minkowski 函数实际上远离原点是平滑的。这允许我们为这些域构建一个平滑的多次谐波定义函数。即使在平衡域的情况下，我们的结果也是新的。