  The interrelations between (upper and lower) Minkowski contents and (upper
and lower) surface area based contents (S-contents) as well as between their
associated dimensions have recently been investigated for general sets in R^d
(cf. [3]). While the upper dimensions always coincide and the upper contents
are bounded by each other, the bounds obtained in [3] suggest that there is
much more flexibility for the lower contents and dimensions.
  We show that this is indeed the case. There are sets whose lower S-dimension
is strictly smaller than their lower Minkowski dimension. More precisely, given
two numbers s, m with 0 < s < m < 1, we construct sets in R^d with lower
S-dimension s+d-1 and lower Minkowski dimension m+d-1. In particular, these
sets are used to demonstrate that the inequalities obtained in [3] regarding
the general relation of these two dimensions are best possible.
