  We study the circumradius of a random section of an $\ell_p$-ellipsoid,
$0<p\le \infty$, and compare it with the minimal circumradius over all sections
with subspaces of the same codimension. Our main result is an upper bound for
random sections, which we prove using techniques from asymptotic geometric
analysis if $1\leq p \leq \infty$ and compressed sensing if $0<p \leq 1$. This
can be interpreted as a bound on the quality of random (Gaussian) information
for the recovery of vectors from an $\ell_p$-ellipsoid for which the radius of
optimal information is given by the Gelfand numbers of a diagonal operator. In
the case where the semiaxes decay polynomially and $1\le p\le \infty$, we
conjecture that, as the amount of information increases, the radius of random
information either decays like the radius of optimal information or is bounded
from below by a constant, depending on whether the exponent of decay is larger
than the critical value $1-\frac{1}{p}$ or not. If $1\leq p\leq 2$, we prove
this conjecture by providing a matching lower bound. This extends the recent
work of Hinrichs et al. [Random sections of ellipsoids and the power of random
information, Trans. Amer. Math. Soc., 2021+] for the case $p=2$.
