Accurate estimation of tail probabilities of projections of high-dimensional probability measures is of relevance in high-dimensional statistics and asymptotic geometric analysis. For fixed $p in (1,infty)$, let $(X^{(n,p)})$ and $(theta^n)$ be independent sequences of random vectors with $theta^n$ distributed according to the normalized cone measure on the unit $ell_2^n$ sphere, and $X^{(n,p)}$ distributed according to the normalized cone measure on the unit $ell_p^n$ sphere. For almost every sequence of projection directions $(theta^n)$, (quenched) sharp large deviation estimates are established for suitably normalized (scalar) projections of $X^{n,p}$ onto $theta^n$, that are asymptotically exact (as the dimension $n$ tends to infinity). Furthermore, the case when $(X^{(n,p)})$ is replaced with $(mathscr{X}^{(n,p)})$, where $mathscr{X}^{(n,p)}$ is distributed according to the uniform (or normalized volume) measure on the unit $ell_p^n$ ball, is also considered. In both cases, in contrast to the (quenched) large deviation rate function, the prefactor exhibits a dependence on the projection directions $(theta^n)$ that encodes geometric information. Moreover, although the (quenched) large deviation rate functions for the sequences of random projections of $(X^{(n,p)})$ and $(mathscr{X}^{(n,p)})$ are known to coincide, it is shown that the prefactor distinguishes between these two cases. The results on the one hand provide quantitative estimates of tail probabilities of random projections of $ell_p^n$ balls and spheres, valid for finite $n$, generalizing previous results due to Gantert, Kim and Ramanan, and on the other hand, generalize classical sharp large deviation estimates in the spirit of Bahadur and Ranga Rao to a geometric setting.
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