Method of Successive Projections for Finding a Common Point of Sets in Metric Spaces

Many problems in applied mathematics can be abstracted into finding a common point of a finite collection of sets. If all the sets are closed and convex in a Hilbert space, the method of successive projections (MOSP) has been shown to converge to a solution point, i.e., a point in the intersection of the sets. These assumptions are however not suitable for a broad class of problems. In this paper, we generalize the MOSP to collections of approximately compact sets in metric spaces. We first define a sequence of successive projections (SOSP) in such a context and then proceed to establish conditions for the convergence of a SOSP to a solution point. Finally, we demonstrate an application of the method to digital signal restoration.