  We consider a Gaussian field $X = \{X_t, t \in T\}$ 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 $\cal P$ of $X_t$ is independent of $t.$ We
consider the asymptotic behavior of closed convex hulls $$ W_n = \conv \{X_t, t
\in T_n\} $$ where $(T_n)$ is an increasing sequence of subsets of $T$ and we
show that under some conditions of the weak dependence with probability 1 $$
\lim_{n\rightarrow \infty} \frac{1}{b_n}\,W_n = {\cal E} $$ (in the sense of
Hausdorff distance), where the limit shape ${\cal E}$ is the concentration
ellipsoid of $\cal P.$
  The asymptotic behavior of the mathematical expectations $Ef(W_n),$ where $f$
is an homogeneous function is also studied.
