  We report on the discovery of a surprising observed correlation between the
slope of the low-mass stellar global mass function (GMF) of globular clusters
(GCs) and their central concentration parameter c = log(r_t/r_c), i.e. the
logarithmic ratio of tidal and core radii. This result is based on the analysis
of a sample of twenty Galactic GCs, with solid GMF measurements from deep HST
or VLT data, representative of the entire population of Milky Way GCs. While
all high-concentration clusters in the sample have a steep GMF,
low-concentration clusters tend to have a flatter GMF implying that they have
lost many stars via evaporation or tidal stripping. No GCs are found with a
flat GMF and high central concentration. This finding appears
counter-intuitive, since the same two-body relaxation mechanism that causes
stars to evaporate and the cluster to eventually dissolve should also lead to
higher central density and possibly core-collapse. Therefore, severely depleted
GCs should be in a post core-collapse state, contrary to what is suggested by
their low concentration. Several hypotheses can be put forth to explain the
observed trend, none of which however seems completely satisfactory. It is
likely that GCs with a flat GMF have a much denser and smaller core than
suggested by their surface brightness profile and may well be undergoing
collapse at present. It is, therefore, likely that the number of post
core-collapse clusters in the Galaxy is much larger than thought so far.
