An official website of the United States government
We prove that the class of reflexive asymptotic-$$c_{0}$$ Banach spaces is coarsely rigid, meaning that if a Banach space $$X$$ coarsely embeds into a reflexive asymptotic-$$c_{0}$$ space $$Y$$, then $$X$$ is also reflexive and asymptotic-$$c_{0}$$. In order to achieve this result, we provide a purely metric characterization of this class of Banach spaces. This metric characterization takes the form of a concentration inequality for Lipschitz maps on the Hamming graphs, which is rigid under coarse embeddings. Using an example of a quasi-reflexive asymptotic-$$c_{0}$$ space, we show that this concentration inequality is not equivalent to the non-equi-coarse embeddability of the Hamming graphs.
more »
« less
This content will become publicly available on December 1, 2025
On Stability of Metric Spaces and Kalton’s Property Q
ABSTRACT The first named author introduced the notion of upper stability for metric spaces in F. Baudier, Barycentric gluing and geometry of stable metrics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 no. 1, (2022), 48 as a relaxation of stability. The motivation was a search for a new invariant to distinguish the class of reflexive Banach spaces from stable metric spaces in the coarse and uniform category. In this paper we show that property Q does in fact imply upper stability. We also provide a direct proof of the fact that reflexive spaces are upper stable by relating the latter notion to the asymptotic structure of Banach spaces.
more »
« less
- PAR ID:
- 10581255
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- The Quarterly Journal of Mathematics
- Volume:
- 75
- Issue:
- 4
- ISSN:
- 0033-5606
- Page Range / eLocation ID:
- 1375 to 1391
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
In this paper we present new proofs of the non-embeddability of countably branching trees into Banach spaces satisfying property beta_p and of countably branching diamonds into Banach spaces which are l_p-asymptotic midpoint uniformly convex (p-AMUC) for p>1. These proofs are entirely metric in nature and are inspired by previous work of Jiří Matoušek. In addition, using this metric method, we succeed in extending these results to metric spaces satisfying certain embedding obstruction inequalities. Finally, we give Tessera-type lower bounds on the compression for a class of Lipschitz embeddings of the countably branching trees into Banach spaces containing l_p-asymptotic models for p>=1.more » « less
-
Abstract We establish that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided d ⩾ 4. We further present an extension of this result to configurations that are the product of two non-degenerate simplices; specifically we show that if Δ k 1 and Δ k 2 are two fixed non-degenerate simplices of k 1 + 1 and k 2 + 1 points respectively, then any subset of ℝ d of positive upper Banach density with d ⩾ k 1 + k 2 + 6 will necessarily contain an isometric copy of all sufficiently large dilates of Δ k 1 × Δ k 2 . A new direct proof of the fact that any subset of ℝ d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed non-degenerate simplex of k + 1 points provided d ⩾ k + 1, a result originally due to Bourgain, is also presented.more » « less
-
null (Ed.)We establish the convergence of the forward-backward splitting algorithm based on Bregman distances for the sum of two monotone operators in reflexive Banach spaces. Even in Euclidean spaces, the convergence of this algorithm has so far been proved only in the case of minimization problems. The proposed framework features Bregman distances that vary over the iterations and a novel assumption on the single-valued operator that captures various properties scattered in the literature. In the minimization setting, we obtain rates that are sharper than existing ones.more » « less
-
Abstract We study a generalization of the classical multidimensional scaling procedure (cMDS) which is applicable in the setting of metric measure spaces. Metric measure spaces can be seen as natural ‘continuous limits’ of finite data sets. Given a metric measure space $${\mathcal{X}} = (X,d_{X},\mu _{X})$$, the generalized cMDS procedure involves studying an operator which may have infinite rank, a possibility which leads to studying its traceability. We establish that several continuous exemplar metric measure spaces such as spheres and tori (both with their respective geodesic metrics) induce traceable cMDS operators, a fact which allows us to obtain the complete characterization of the metrics induced by their resulting cMDS embeddings. To complement this, we also exhibit a metric measure space whose associated cMDS operator is not traceable. Finally, we establish the stability of the generalized cMDS method with respect to the Gromov–Wasserstein distance.more » « less
