Łojasiewicz-Siciak condition for the pluricomplex Green function

Volume 92 / 2011

Marta Kosek Banach Center Publications 92 (2011), 197-203 MSC: Primary 31-02, 32-02; Secondary 32U35. DOI: 10.4064/bc92-0-13


A compact set $K\subset\mathbb{C}^N$ satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants $B,\beta>0$ such that $$\def\dist{\mathop{\rm dist}} V_K(z)\geq B(\dist(z,K))^\beta\quad {\rm if}\quad \dist(z,K)\leq 1. \tag*{(LS)} $$ Here $V_K$ denotes the pluricomplex Green function of the set $K$. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).


  • Marta KosekInstitute of Mathematics
    Jagiellonian University
    ul. Łojasiewicza 6
    30-348 Kraków, Poland

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