Testing surface area with arbitrary accuracy

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 $κ_n S$. Here, $κ_n$ is a dimension-dependent constant which is strictly larger than 1 if $n \ge 2$, and grows to $4/π$ as $n \to \infty$. We give an improved analysis of Kothari et al.'s algorithm. In doing so, we replace the constant $κ_n$ with $1 + η$ for $η> 0$ arbitrary. We also extend the algorithm to more general measures on Riemannian manifolds.