  The requirement that packings of hard particles, arguably the simplest
structural glass, cannot be compressed by rearranging their network of contacts
is shown to yield a new constraint on their microscopic structure. This
constraint takes the form a bound between the distribution of contact forces
P(f) and the pair distribution function g(r): if P(f) \sim f^{\theta} and g(r)
\sim (r-{\sigma})^(-{\gamma}), where {\sigma} is the particle diameter, one
finds that {\gamma} \geq 1/(2+{\theta}). This bound plays a role similar to
those found in some glassy materials with long-range interactions, such as the
Coulomb gap in Anderson insulators or the distribution of local fields in
mean-field spin glasses. There is ground to believe that this bound is
saturated, offering an explanation for the presence of avalanches of
rearrangements with power-law statistics observed in packings.
