  The complete spectrum of the endstates - naked singularities, or blackholes -
of gravitational collapse is analyzed for a wide class of $N$-dimensional
spacetimes in spherical symmetry, which includes and generalizes the dust
solutions and the case of vanishing radial stresses. The final fate of the
collapse is shown to be fully determined by the local behavior of a single
scalar function and by the dimension $N$ of the spacetime. In particular, the
``critical'' behavior of the N=4 spacetimes, where a sort of phase transition
from black hole to naked singularity can occur, is still present if N=5 but
does not occur if $N > 5$, independently from the initial data of the collapse.
Physically, the results turn out to be related to the kinematical properties of
the considered solutions.
