  We provide an algorithm of constructing a rectifiable curve between two
sufficiently close points of a proximally smooth set in a uniformly convex and
uniformly smooth Banach space. Our algorithm returns a reasonably short curve
between two sufficiently close points of a proximally smooth set, is iterative
and uses a certain modification of the metric projection. We estimate the
length of a constructed curve and its deviation from the segment with the same
endpoints. These estimates coincide up to a constant factor with those for the
geodesics in a proximally smooth set in a Hilbert space.
