Continuity of Metric Projection, Pólya Algorithm, Strict Best Approximation, and Tubularity of Convex Sets

The notion of tubularity of a convex subset, K, of l∞ (n) was originally introduced to study the convergence of the Pólya algorithm. It is shown in the present paper that this geometric condition provides a characterization of thosed closed convex sets onto which the set-valued metric projection is continuous. In the development of this result, Rice′s strict best approximation is characterized in three new ways, and is shown, assuming tubularity of K, to be a continuous selection. The class of sets on which the Pólya algorithm is known to converge is enlarged to include all closed convex totally tubular sets. Tubularity is shown to be related to the (P)-sets introduced, in a study of the metric projection, by Brown and Wegmann.