Despite its successes in explaining the richness and variety of the
large-scale structures in the Universe, the Cold Dark Matter (CDM)
paradigm is known to show some discrepancy with observations on the
smallest (~ kiloparsec) scales. In collaboration with Christian Marinoni
(CPT, Marseille) I have formulated gravitational suppression (GraS)
hypothesis, namely, that the effective gravitational interaction
betweendark matter and visible matter is suppressed on subgalactic
(~ kpc) scales. Such a simple phenomenological hypothesis, while
consistent with the observed large scale structures and CMB
anisotropies, solves the cusp problem of the velocity curves of low-
surface brightness galaxies and regulates the baryon fraction inside dark matter halos of different sizes. We have also considered the velocity scales measurable in dwarf spheroidal galaxies and found that, compared to Newtonian predictions, GraS provides a better description of data, when realistic dark matter density profiles are considered. Later studies on different sets of galaxies, while confirming the improvement of the velocity curves fitting, estimate as ``unphysical" the mass parameter fitted with GraS.