Strange shadows of ${\mathrm{\ell }}_p$-balls

We prove a large deviations principle for orthogonal projections of the unit ball ${\mathbb{B}}_p^n$ of ${\mathrm{\ell }}_p^n$ onto a random $k$-dimensional linear subspace of ${\mathbb{R}}^n$ as $n\to \mathrm{\infty }$ in the case $2<p\le \mathrm{\infty }$ and for the intersection of ${\mathbb{B}}_p^n$ with a random $k$-dimensional subspace in the case $1\le p<2$. The corresponding rate function is finite only on $L_q$-zonoids and their duals, respectively, and given in terms of the maximum entropy over suitable measures generating the $L_q$-zonoid, where $\frac{1}{p}+\frac{1}{q}=1$. In particular, we obtain that the renormalized projections/sections almost surely tend to a $k$-dimensional Euclidean ball of certain radius. Moreover, we identify the asymptotic probability that the random orthogonal projection remains within a ball of smaller radius. As a byproduct we obtain an interesting inequality for the Gamma function.