Reverse Euclidean and Gaussian isoperimetric inequalities for parallel sets with applications

• URL: http://arxiv.org/abs/2006.09568v2
• Date: Fri, 21 Aug 2020 02:47:10 GMT
• Title: Reverse Euclidean and Gaussian isoperimetric inequalities for parallel sets with applications
• Authors: Varun Jog
• Abstract summary: We show that the surface area of an $r$-parallel set in $mathbb Rd$ with volume at most $V$ is upper-bounded by $eTheta(d)V/r$, whereas its Gaussian surface area is upper-bounded by $max(eTheta(d), eTheta(d)/r)$. 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
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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