Testing surface area with arbitrary accuracy

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 $4\mathrm{/}\pi$ as $n\to \mathrm{\infty }$. 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.