Rectifiable curves in proximally smooth sets

TL;DR

This paper presents an algorithm for constructing short, rectifiable curves between close points in proximally smooth sets within Banach spaces, extending known results from Hilbert spaces and providing length and deviation estimates.

Contribution

It introduces an iterative algorithm using a modified metric projection to construct rectifiable curves in proximally smooth sets in Banach spaces, with length and deviation estimates.

Findings

Algorithm produces reasonably short curves between close points

Length and deviation estimates match those in Hilbert spaces up to a constant

Applicable in uniformly convex and smooth Banach spaces

Abstract

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.