Around some extremal problems for multivariate polynomials

Let E be a compact subset of C-N and V(E )be the pluricomplex Green's function of E . The H & ouml;lder continuity property, HCP for short, is one of the most interesting features of V-E . By means of a radial modification of V-E , we give some equivalent conditions to HCP connected with the Plesniak 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.