  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.
