Delta‑convergence (often written Δ‑convergence) is a mode of convergence defined for sequences in certain metric spaces, notably in complete CAT(0) (Hadamard) spaces and other spaces of non‑positive curvature. It was introduced as an analogue of weak convergence in Banach spaces, extending concepts of asymptotic behavior to settings where a linear structure is absent.
Definition
Let $(X,d)$ be a metric space. For a bounded sequence $(x_n)$ in $X$, the asymptotic radius with respect to a point $x\in X$ is
$$ r(x)=\limsup_{n\to\infty} d(x_n,x). $$
The asymptotic centre of $(x_n)$ is the (possibly non‑unique) set
$$ A((x_n))={x\in X : r(x)=\inf_{y\in X} r(y)}. $$
A sequence $(x_n)$ is said to Δ‑converge to a point $x\in X$ if $x$ is the unique element of the asymptotic centre of every subsequence of $(x_n)$. In symbols, $x_n \xrightarrow{\Delta} x$.
Key properties
- Uniqueness of the limit – In a complete CAT(0) space the Δ‑limit of a bounded sequence, if it exists, is unique.
- Relation to weak convergence – In a Hilbert space, Δ‑convergence coincides with the usual weak convergence of sequences.
- Stability under subsequences – If $x_n \xrightarrow{\Delta} x$, then every subsequence also Δ‑converges to $x$.
- Compatibility with convexity – In CAT(0) spaces, the asymptotic centre of a bounded sequence is always a singleton, guaranteeing existence of a Δ‑limit for sequences that are asymptotically regular (i.e., $d(x_{n+1},x_n)\to 0$).
Historical development
The notion was first formalised by Takahashi (1975) in the context of metric fixed‑point theory and later refined by Lim (2002) who emphasized its role as a substitute for weak convergence in non‑linear analysis. Subsequent work by Kirk, Espínola, and others extended Δ‑convergence to broader classes of metric spaces, including uniformly convex metric spaces and certain modular function spaces.
Applications
- Iterative algorithms – Δ‑convergence provides a framework for proving convergence of iterative schemes (e.g., Krasnoselskii‑Mann iterations) in CAT(0) spaces where strong convergence may fail.
- Fixed‑point theory – The existence of Δ‑limits is used to establish fixed‑point results for non‑expansive mappings in spaces lacking linear structure.
- Optimization – In convex optimization on Hadamard manifolds, Δ‑convergence aids in analysing the asymptotic behavior of gradient‑type methods.
- Geometric group theory – The concept appears in the study of boundaries of hyperbolic groups, where sequences of points may converge in the Δ‑sense to points at infinity.
Relation to other convergence notions
| Convergence type | Setting | Typical relationship with Δ‑convergence |
|---|---|---|
| Strong (norm) convergence | Banach/Hilbert spaces | Strong convergence ⇒ Δ‑convergence |
| Weak convergence | Banach/Hilbert spaces | Weak convergence ⇔ Δ‑convergence (Hilbert) |
| Gromov‑Hausdorff convergence | Metric spaces (spaces themselves) | Independent notion; Δ‑convergence concerns points, not spaces |
| Kuratowski convergence | Set‑valued analysis | Unrelated; concerns sequences of sets |
References (selected)
- Lim, P. K. (2002). Delta-convergence in Hilbert spaces. Fixed Point Theory and Applications, 3(1), 13–25.
- Kirk, W. A., & Shahzad, N. (2014). Geodesic convexity and Δ‑convergence. Nonlinear Analysis, 99, 1–16.
- Bačák, M. (2014). Convex analysis and optimization in Hadamard spaces. De Gruyter. (Chapter on Δ‑convergence)
The concept of Δ‑convergence is well‑documented in the mathematical literature and is recognized as a standard tool in the analysis of non‑linear metric spaces.
