  Randomized smoothing, a method to certify a classifier's decision on an input
is invariant under adversarial noise, offers attractive advantages over other
certification methods. It operates in a black-box and so certification is not
constrained by the size of the classifier's architecture. Here, we extend the
work of Li et al. \cite{li2018second}, studying how the choice of divergence
between smoothing measures affects the final robustness guarantee, and how the
choice of smoothing measure itself can lead to guarantees in differing threat
models. To this end, we develop a method to certify robustness against any
$\ell_p$ ($p\in\mathbb{N}_{>0}$) minimized adversarial perturbation. We then
demonstrate a negative result, that randomized smoothing suffers from the curse
of dimensionality; as $p$ increases, the effective radius around an input one
can certify vanishes.
