  We investigate identifying the boundary of a domain from sample points in the
domain. We introduce new estimators for the normal vector to the boundary,
distance of a point to the boundary, and a test for whether a point lies within
a boundary strip. The estimators can be efficiently computed and are more
accurate than the ones present in the literature. We provide rigorous error
estimates for the estimators. Furthermore we use the detected boundary points
to solve boundary-value problems for PDE on point clouds. We prove error
estimates for the Laplace and eikonal equations on point clouds. Finally we
provide a range of numerical experiments illustrating the performance of our
boundary estimators, applications to PDE on point clouds, and tests on image
data sets.
