Around some extremal problems for multivariate polynomials

Let E be a compact subset of CN and VE be the pluricomplex Green’s function of E. The Hölder continuity property, HCP for short, is one of the most interesting features of VE . By means of a radial modification of VE , we give some equivalent conditions to HCP connected with the Ple ́sniak property and the Markov inequality for polynomials. Moreover, we consider a capacity, a Chebyshev constant and a transfinite diameter with respect to a fixed norm on the space of polynomials of N variables. We prove that this capacity is not greater than a corresponding Chebyshev constant. One section is devoted to economisation procedure of approximation by telescoping series.