Reverse Euclidean and Gaussian isoperimetric inequalities for parallel sets with applications

Abstract

The r-parallel set of a measurable set A ⊆ 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 R d with volume at most V is upper-bounded by e Θ(d) V /r. We also show that the Gaussian surface area of any r-parallel set in R d is upper-bounded by max(e Θ(d) , e Θ(d) /r). 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.

Reverse isoperimetric inequalities for parallel sets

“…In [2] Jog considered reverse isoperimetric inequalities for parallel sets. He proved that for any compact set…”

“…In [2] also the Gaussian case was treated. The upper Gaussian surface area of a measurable set is defined as…”