  We compare and combine two approaches that have been recently introduced by
Dafnis and Paouris [DP] and by Klartag and Milman [KM] with the aim of
providing bounds for the isotropic constants of convex bodies. By defining a
new hereditary parameter for all isotropic log-concave measures, we are able to
show that the method in [KM], and the apparently stronger conclusions it leads
to, can be extended in the full range of the 'weaker' assumptions of [DP]. The
new parameter we define is related to the highest dimension k\leq n-1 in which
one can always find marginals of an n-dimensional isotropic measure which have
bounded isotropic constant.
