  This paper develops a unified framework for estimating the volume of a set in
$\mathbb{R}^d$ based on observations of points uniformly distributed over the
set. The framework applies to all classes of sets satisfying one simple axiom:
a class is assumed to be intersection stable. No further hypotheses on the
boundary of the set are imposed; in particular, the convex sets and the
so-called weakly-convex sets are covered by the framework. The approach rests
upon a homogeneous Poisson point process model. We introduce the so-called
wrapping hull, a generalization of the convex hull, and prove that it is a
sufficient and complete statistic. The proposed estimator of the volume is
simply the volume of the wrapping hull scaled with an appropriate factor. It is
shown to be consistent for all classes of sets satisfying the axiom and mimics
an unbiased estimator with uniformly minimal variance. The construction and
proofs hinge upon an interplay between probabilistic and geometric arguments.
The tractability of the framework is numerically confirmed in a variety of
examples.
