Best choice from the planar Poisson process

Various best-choice problems related to the planar homogeneous Poisson process in a finite or semi-infinite rectangle are studied. The analysis is largely based on the properties of the one-dimensional box-area process associated with the sequence of records. We prove a series of distributional identities involving exponential and uniform random variables, and give a resolution to the Petruccelli-Porosinski-Samuels paradox on the coincidence of asymptotic values in certain discrete-time optimal stopping problems.