  The $r$-parallel set of a measurable set $A \subseteq \mathbb R^d$ is the set
of all points whose distance from $A$ is at most $r$. In this paper, we show
that the surface area of an $r$-parallel set in $\mathbb R^d$ with volume at
most $V$ is upper-bounded by $e^{\Theta(d)}V/r$, whereas its Gaussian surface
area is upper-bounded by $\max(e^{\Theta(d)}, e^{\Theta(d)}/r)$. We also derive
a reverse form of the Brunn-Minkowski inequality for $r$-parallel sets, and as
an aside a reverse entropy power inequality for Gaussian-smoothed random
variables. We apply our results to two problems in theoretical machine
learning: (1) bounding the computational complexity of learning $r$-parallel
sets under a Gaussian distribution; and (2) bounding the sample complexity of
estimating robust risk, which is a notion of risk in the adversarial machine
learning literature that is analogous to the Bayes risk in hypothesis testing.
