  We investigate the relation between fragility and phase space properties -
such as the distribution of states - in the mean field p-spin model, a solvable
model that has been frequently used in studies of the glass transition. By
direct computation of all the relevant quantities, we find that: i) the
recently observed correlation between fragility and vibrational properties at
low temperature is present in this model; ii) the total number of states is a
decreasing function of fragility, at variance of what is currently believed. We
explain these findings by taking into account the contribution to fragility
coming from the transition paths between different states. Finally, we propose
a geometric picture of the phase space that explains the correlation between
properties of the transition paths, distribution of states and their
vibrational properties. However, our analysis may not apply to strong systems
where inflection points in the configurational entropy as a function of the
temperature are found.
