Rectifiable curves in proximally smooth sets (Q2670984)

Some geometric properties of proximally smooth sets are studied. A proximally smooth set, as defined by Clarke, Stern, and Wolenski, is a set such that the distance function to it from a point of the space is continuously differentiable in some neighbourhood of this set excluding the set itself. A modification of the metric projection operator (a slice-projection) is introduced. It is shown that the slice-projection operator is nonempty in any smooth and uniformly convex Banach space. An algorithm is constructed for producing a rectifiable curve between two sufficiently close points of a proximally smooth set in a uniformly convex and uniformly smooth Banach space with power-law moduli of smoothness and convexity. The algorithm, which returns a reasonably short curve between two sufficiently close points of a proximally smooth set, is iterative and uses our modification of the metric projection. The length of the constructed curve and its deviation from the segment with the same endpoints is estimated. The obtained estimates coincide up to a constant factor with those for the geodesics in a proximally smooth subset of a~Hilbert space.