Reverse Euclidean and Gaussian isoperimetric inequalities for parallel sets with applications

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.