Reverse Lebesgue and Gaussian isoperimetric inequalities for parallel sets with applications
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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.
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