  Various best-choice problems related to the planar homogeneous Poisson
process in finite or semi-infinite rectangle are studied. The analysis is
largely based on 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 resolve the
Petruccelli-Porosinski-Samuels paradox on coincidence of asymptotic values in
certain discrete-time optimal stopping problems.
