  Sharp large deviation results of Bahadur-Ranga Rao type are provided for the
$q$-norm of random vectors distributed on the $\ell _{p}^{n}$-ball
${\mathbb{B}}^{n}_{p}$ according to the cone probability measure or the uniform
distribution for $1 \le q<p < \infty $, thereby furthering previous large
deviation results by Kabluchko, Prochno and Th\"{a}le in the same setting.
These results are then applied to deduce sharp asymptotics for intersection
volumes of different $\ell _{p}^{n}$-balls in the spirit of Schechtman and
Schmuckenschl\"{a}ger, and for the length of the projection of an $\ell
_{p}^{n}$-ball onto a line with uniform random direction. The sharp large
deviation results are proven by providing convenient probabilistic
representations of the $q$-norms, employing local limit theorems to approximate
their densities, and then using geometric results for asymptotic expansions of
Laplace integrals to integrate these densities and derive concrete probability
estimates.
