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Given two disjoint convex polyhedra, we look for a best approximation pairrelative to them, i.e., a pair of points, one in each polyhedron, attaining theminimum distance between the sets. Cheney and Goldstein showed that alternatingprojections onto the two sets, starting from an arbitrary point, generate asequence whose two interlaced subsequences converge to a best approximationpair. We propose a process based on projections onto the half-spaces definingthe two polyhedra, which are more negotiable than projections on the polyhedrathemselves. A central component in the proposed process is theHalpern--Lions--Wittmann--Bauschke algorithm for approaching the projection ofa given point onto a convex set.

Finding a best approximation pair of points for two polyhedra