We present an algorithm for finding a nearest pair of points in two convex
sets of Rn, and therefore, their distance. The algorithm is based on the fi
xed-point theory of nonexpansive operators on a Hilbert space. Its practica
l implementation requires a fast projection algorithm. We introduce such a
procedure for convex polyhedra. This algorithm effects a local search in th
e faces using visibility as a guide for finding the global minimum. After s
tudying the convergence of both algorithms, we detail computer experiments
on polyhedra (projection and distance). In the case of distances, these exp
eriments show a sublinear time complexity relative to the total number of v
ertices. (C) 2000 Elsevier Science Ltd. All rights reserved.