Abstract
Let K(p) be the set of (p, p) closed positive and conic currents on C(n) such that the Lelong number is equal to I at the origine O, the convex subset K(p) is a weakly compact metrizable subspace of the space of currents on C(n). For every current T on C(n), the set of limits of the family ((h(r))* T)r<1 is connex and weakly compact in m. K(p) where m is the Lelong number of T at O. Conversely we prove in this Note that if M is any closed connex subset of K(p) (n - 1 greater-than-or-equal-to p greater-than-or-equal-to 2); there exists a closed positive current T such that M is the limit set of the family ((h(r))* T)r<1. The case p = 1 was first treated by C. O. Kiselman [51, and then refined by the author in collaboration with J. P. Demailly and M. Mouzali [1] to get precise quantitative estimates. In the second part we give a sufficient condition relative to the trace measure for the existence of the tangent cone of a closed positive current of type (p, p) with p = 1 or n - 1. We prove that this condition is not valid for the case 2 less-than-or-equal-to p less-than-or-equal-to n - 2.