1. Let k points z_1, z_2, ... z_k be given in the plane of complex numbers, and let a_1, a_2, ... a_k be nonnegative numbers, for which a1 + a2 + ... + a_k = 1. Then the number, y = a_1*z_1 + a_2*z_2 + ... + a_k*z_k lies in the smallest convex (closed) polygon containing the points z_1, z_2, ... z_k.
Is this still true for infinite sequences {z_n} and {z_n}, provided that \sum a_n = 1 and \sum a_n*z_n = y exists?