Siciak's extremal function via Bernstein and Markov constants for compact sets in CN

The paper is concerned with the best constants in the Bernstein and Markov inequalities on a compact set E subset of C-N. We give some basic properties of these constants and we prove that two extremal-like functions defined in terms of the Bernstein constants are plurisubharmonic and very close to the Siciak extremal function Phi(E). Moreover, we show that one of these extremal-like functions is equal to Phi(E) if E is a nonpluripolar set with lim(n ->infinity) M-n(E)(1/n) = 1 where (0.1) M-n(E) := sup parallel to vertical bar grad P vertical bar parallel to(E)/parallel to P parallel to(E), the supremum is taken over all polynomials P of N variables of total degree at most n and parallel to . parallel to(E) is the uniform norm on E. The above condition is fulfilled e.g. for all regular (in the sense of the continuity of the pluricomplex Green function) compact sets in C-N.