  We study the Vitale zonoid (a convex body associated to a probability
distribution) associated to a non--centered Gaussian vector. We compute an
ellipsoid that contains such zonoid and show that it remains close to it. We
use this result to give new estimates for the expectation of the absolute
determinant of a non--centered Gaussian matrix in terms of mixed volume of
ellipsoids. Finally, exploiting a recent link between random fields and zonoids
uncovered by Stecconi and the author, we apply our results to the study of the
zero set of non--centered Gaussian random fields. We show how these can be
approximated by a suitable centered Gaussian random field. We also give an
explicit asymptotic formula for the concentration of measure in the limit where
the variance goes to zero.
