Sharp large deviation results of Bahadur-Ranga Rao type are provided for the qnorm of random vectors distributed on the n p -ball B n p according to the cone probability measure or the uniform distribution for 1 ≤ q < p < ∞, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different n p -balls in the spirit of Schechtman and Schmuckenschläger, and for the length of the projection of an n p -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.
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