Cyclic projections in Hadamard spaces

TL;DR

This paper demonstrates that cyclic projections in Hadamard spaces can exhibit complex behavior unlike in Hilbert spaces, providing a counterexample to asymptotic regularity in such settings.

Contribution

It constructs the first example of non-asymptotic regular cyclic projections in Hadamard spaces, addressing a problem posed by Bačák.

Findings

Cyclic projections can behave more complex in Hadamard spaces than in Hilbert spaces.

Provided a specific example of convex subsets with non-asymptotic regular cyclic projections.

Shows that cyclic projections need not always converge in Hadamard spaces.

Abstract

We show that cyclic products of projections onto convex subsets of Hadamard spaces can behave in a more complicated way than in Hilbert spaces, resolving a problem formulated by Miroslav Ba\v{c}\'ak. Namely, we construct an example of convex subsets in a Hadamard space such that the corresponding cyclic product of projections is not asymptotically regular.