With growing dimension, a typical random projection of the
$\ell_p^n$-ball onto a subspace of fixed dimension tends to a Euclidean
ball of some fixed radius. This is related to the strong law of large
numbers of the $p^*$-sum of independent and identically distributed line
segments, where $p^*$ is the conjugate index. It is thus not surprising
that $L_{p^*}$-zonoids appear as shadows and the typical shadow of the
$\ell_p^n$-ball is close to the above Euclidean ball. We are interested
in shadows which are strange, meaning that they occur with probability
exponentially decaying with some rate. This is formalized by a large
deviations principle in the space of convex bodies equipped with
Hausdorff distance in the case of $p>2$. Building on work of Kim and
Ramanan, we identify the rate of decay via the entropy of representing
measures of the corresponding $L_{p^*}$-zonoid. Via duality we obtain a
result for random sections. Based on joint work with Zakhar Kabluchko.