We consider a centered Gaussian random field X = {X _t: t ∈ T} with values in a Banach space B defined on a parametric set T equal to ℝ^m or ℤ^m. It is supposed that the distribution P of X _t is independent of t. We consider the asymptotic behavior of closed convex hulls W _n = conv{X _t: t ∈ T _n}, where (T _n) is an increasing sequence of subsets of T. We show that under some conditions of weak dependence for the random field under consideration and some sequence (b _n)_n≥1 with probability 1, (Formula presented.) (in the sense of Hausdorff distance), where the limit set [InlineMediaObject not available: see fulltext.] is the concentration ellipsoid of P. The asymptotic behavior of the mathematical expectations Ef(W _n), where f is some function, is also studied.