Minimax Estimation of the Volume of a Set under the Rolling Ball Condition

We consider the problem of estimating the volume of a compact domain in a Euclidean space based on a uniform sample from the domain. We assume that the domain has a boundary with positive reach. We propose a data splitting approach to correct the bias of the plug-in estimator based on the sample α-convex hull. We show that this simple estimator achieves a minimax lower bound that we derive. Some numerical experiments corroborate our theoretical findings.