Convergences of Alternating Projections in CAT($\kappa$) Spaces

TL;DR

This paper investigates the behavior of alternating projection sequences in CAT(κ) spaces, proving their convergence properties and establishing conditions for strong convergence in these non-linear geometric spaces.

Contribution

The paper extends convergence results of alternating projections to CAT(κ) spaces with positive curvature, including asymptotic regularity and strong convergence under specific conditions.

Findings

Asymptotic regularity of projection sequences established

Δ-convergence of sequences proved in CAT(κ) spaces

Strong convergence shown under regularity or compactness assumptions

Abstract

We establish the asymptotic regularity and the $\Delta$-convergence of the sequence constructed by the alternating projections to closed convex sets in a CAT($\kappa$) space with $\kappa > 0$. Furthermore, the strong convergence of the alternating von Neumann sequence is presented under certain regularity or compactness.