Let
$K$
be a convex body in
${{\mathbf{E}}^{d}}$
and denote by
${{C}_{n}}$
the set of centroids of
$n$
non-overlapping translates of
$K$
. For
$\varrho \,>\,0$
, assume that the parallel body conv
${{C}_{n}}\,+\,\varrho K$
of conv
${{C}_{n}}$
has minimal volume. The notion of parametric density (see [21]) provides a bridge between finite and infinite packings (see [4] or [14]). It is known that there exists a maximal
${{\varrho }_{s}}(K)\,\ge \,1/(32{{d}^{2}})$
such that conv
${{C}_{n}}$
is a segment for
$\varrho \,<\,{{\varrho }_{s}}$
(see [5]). We prove the existence of a minimal
${{\varrho }_{c}}(K)\,\le \,d\,+\,1$
such that if
$\varrho \,>\,{{\varrho }_{c}}$
and
$n$
is large then the shape of conv
${{C}_{n}}$
can not be too far from the shape of
$K$
. For
$d\,=\,2$
, we verify that
${{\varrho }_{s\,}}\,=\,{{\varrho }_{c}}$
. For
$d\,\ge \,3$
, we present the first example of a convex body with known
${{\varrho }_{s}}$
and
${{\varrho }_{c}}$
; namely, we have
${{\varrho }_{s}}\,=\,{{\varrho }_{c}}\,=\,1$
for the parallelotope.