  We introduce a hull operator on Poisson point processes, the easiest example
being the convex hull of the support of a point process in Euclidean space.
Assuming that the intensity measure of the process is known on the set
generated by the hull operator, we discuss estimation of an expected linear
statistic built on the Poisson process. In special cases, our general scheme
yields an estimator of the volume of a convex body or an estimator of an
integral of a H\"older function. We show that the estimation error is given by
the Kabanov--Skorohod integral with respect to the underlying Poisson process.
A crucial ingredient of our approach is a spatial strong Markov property of the
underlying Poisson process with respect to the hull. We derive the rate of
normal convergence for the estimation error, and illustrate it on an
application to estimators of integrals of a H\"older function. We also discuss
estimation of higher order symmetric statistics.
