  Let $p\in[1,\infty]$. Consider the projection of a uniform random vector from
a suitably normalized $\ell^p$ ball in $\mathbb{R}^n$ onto an independent
random vector from the unit sphere. We show that sequences of such random
projections, when suitably normalized, satisfy a large deviation principle
(LDP) as the dimension $n$ goes to $\infty$, which can be viewed as an annealed
LDP. We also establish a quenched LDP (conditioned on a fixed sequence of
projection directions) and show that for $p\in(1,\infty]$ (but not for $p=1$),
the corresponding rate function is "universal", in the sense that it coincides
for "almost every" sequence of projection directions. We also analyze some
exceptional sequences of directions in the "measure zero" set, including the
directions corresponding to the classical Cram\'er's theorem, and show that
those directions yield LDPs with rate functions that are distinct from the
universal rate function of the quenched LDP. Lastly, we identify a variational
formula that relates the annealed and quenched LDPs, and analyze the minimizer
of this variational formula. These large deviation results complement the
central limit theorem for convex sets, specialized to the case of sequences of
$\ell^p$ balls.
