Testing surface area with arbitrary accuracy

TL;DR

This paper improves the analysis of an algorithm for testing the surface area of sets in high-dimensional spaces, achieving arbitrary accuracy and extending to Riemannian manifolds.

Contribution

It refines the constant factor in the surface area testing algorithm to be arbitrarily close to 1 and extends the method to general measures on Riemannian manifolds.

Findings

Constant factor improved to 1 + η for any η > 0

Algorithm extended to Riemannian manifolds

Enhanced analysis of surface area testing algorithm

Abstract

Recently, Kothari et al.\ gave an algorithm for testing the surface area of an arbitrary set $A \subset [0, 1]^n$. Specifically, they gave a randomized algorithm such that if $A$'s surface area is less than $S$ then the algorithm will accept with high probability, and if the algorithm accepts with high probability then there is some perturbation of $A$ with surface area at most $\kappa_n S$. Here, $\kappa_n$ is a dimension-dependent constant which is strictly larger than 1 if $n \ge 2$, and grows to $4/\pi$ as $n \to \infty$. We give an improved analysis of Kothari et al.'s algorithm. In doing so, we replace the constant $\kappa_n$ with $1 + \eta$ for $\eta > 0$ arbitrary. We also extend the algorithm to more general measures on Riemannian manifolds.