Chebyshev sets in geodesic spaces

On Approximation Methods in Some Geodesic Spaces without the Nice Projection Property

“…In CAT(1) spaces, it is known that the metric projection is a Lipschitz mapping although it may be expansive [18]. The following lemma can be deduced from the proof of [17,Theorem 4.1] together with the above lemma.…”

Local Linear Convergence of Alternating Projections in Metric Spaces with Bounded Curvature

“…If C is additionally complete in the induced metric, then x always has a nearest point in C, and hence P C (x) is a singleton. We refer to [15,5] for a more thorough discussion on the behavior of the metric projection in CAT(κ) spaces.…”

“…In order to elaborate a more coherent theory, in the present work we introduce the concept of subdifferential using normal cones. Since the notion of normal cone can be based on the metric projection whose properties are rich enough in Alexandrov spaces of curvature bounded above (see [5]), we consider this context and briefly discuss in Section 2 some of its fundamental properties, together with other notions used in what follows. However, we also impose local compactness in our framework in order to obtain an analogue of the supporting hyperplane theorem from finite-dimensional Hilbert spaces, which allows us to establish the existence of subgradients at a continuity point of a convex function.…”

Local linear convergence of alternating projections in metric spaces with bounded curvature

“…A systematic study of properties of Chebyshev sets (i.e. sets where the metric projection is a singleton for all points in the space) in Alexandrov spaces is carried out in [3]. CBB(κ) spaces (or spaces that have curvature bounded below by κ in the sense of Alexandrov) are defined in a similar way to CAT(κ) spaces using in this case the reverse of the CAT(κ) inequality.…”