  We derive a microscopic criterion for the stability of hard sphere
configurations, and we show empirically that this criterion is marginally
satisfied in the glass. This observation supports a geometric interpretation
for the initial rapid rise of viscosity with packing fraction, or
pre-vitrification. It also implies that barely stable soft modes characterize
the glass structure, whose spatial extension is estimated. We show that both
the short-term dynamics and activation processes occur mostly along those soft
modes, and we study some implications of these observations. This article
synthesizes new and previous results [C. Brito and M. Wyart, Euro. Phys.
Letters, {\bf 76}, 149-155, (2006) and C. Brito and M. Wyart, J. Stat. Mech.,
L08003 (2007) ] in a unified view.
