  The paper provides a description of the large deviation behavior for the
Euclidean norm of projections of $\ell_p^n$-balls to high-dimensional random
subspaces. More precisely, for each integer $n\geq 1$, let
$k_n\in\{1,\ldots,n-1\}$, $E^{(n)}$ be a uniform random $k_n$-dimensional
subspace of $\mathbb R^n$ and $X^{(n)}$ be a random point that is uniformly
distributed in the $\ell_p^n$-ball of $\mathbb R^n$ for some $p\in[1,\infty]$.
Then the Euclidean norms $\|P_{E^{(n)}}X^{(n)}\|_2$ of the orthogonal
projections are shown to satisfy a large deviation principle as the space
dimension $n$ tends to infinity. Its speed and rate function are identified,
making thereby visible how they depend on $p$ and the growth of the sequence of
subspace dimensions $k_n$. As a key tool we prove a probabilistic
representation of $\|P_{E^{(n)}}X^{(n)}\|_2$ which allows us to separate the
influence of the parameter $p$ and the subspace dimension $k_n$.
