  We investigate the structure of metastable states in self-generated Coulomb
glasses. In dramatic contrast to disordered electron glasses, we find that
these states lack marginal stability. Such absence of marginal stability is
reflected by the suppression of the single-particle density of states into an
exponentially soft gap of the form $g(\epsilon) \sim e^{-V / \xi |\epsilon|}$.
To analytically explain this behavior, we extend the stability criterion of
Efros and Shklovskii to incorporate local charge correlations, in qualitative
agreement with our numerical findings. Our work suggests the existence of a new
class of self-generated glasses dominated by strong geometric frustration.
