Core Concepts

The article establishes conditions for the existence of continuous extensions of Lipschitz maps and monotone maps in Hilbert spaces, particularly focusing on preserving the uniform distance between the extended map and a given reference map.

Abstract

**Bibliographic Information:**Ciosmak, K. J. (2024). Continuity of extensions of Lipschitz maps and of monotone maps. arXiv preprint arXiv:2402.14699v2.**Research Objective:**The paper investigates the conditions under which a given map defined on a subset of a Hilbert space allows for a distance-preserving Lipschitz or monotone extension of any Lipschitz or monotone map defined on a subset of its domain.**Methodology:**The study employs theoretical analysis and utilizes tools from functional analysis, including the Kirszbraun theorem, the Helly theorem, the Banach-Alaoglu theorem, and the Kuratowski–Zorn lemma. The concept of Kirszbraun functions and properties of leaves of 1-Lipschitz maps are central to the proofs.**Key Findings:**- The paper proves that a specific set of inequalities concerning a map
*v*defined on a subset*X*of a Hilbert space is sufficient to guarantee the existence of a distance-preserving Lipschitz extension for any Lipschitz map defined on a subset of*X*. - This condition is also proven to be necessary when the dimension of the codomain is less than or equal to 3 or when
*X*is convex. - Similar results are established for monotone maps, demonstrating the existence of monotone extensions under analogous conditions.
- The paper also explores the extension properties of maps of 1-semi-bounded strain, a class of functions relevant to the study of Michell trusses.

- The paper proves that a specific set of inequalities concerning a map
**Main Conclusions:**The paper provides a significant contribution to the understanding of Lipschitz and monotone map extensions in Hilbert spaces. The established conditions for distance-preserving extensions have implications for various fields, including variational problems, optimal transport, and geometric analysis.**Significance:**The findings are particularly relevant to the study of multi-dimensional localization in optimal transport problems, addressing a conjecture by Klartag and providing insights into the limitations of the mass balance condition in higher dimensions.**Limitations and Future Research:**The equivalence between the existence of distance-preserving extensions and the established condition is proven only for specific cases (dimension of codomain less than or equal to 3 or convexity of the domain). Further research could explore the necessity of the condition in more general settings. Additionally, investigating the continuity properties of extensions for maps of 1-semi-bounded strain could be a promising direction.

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The paper focuses on maps with values in R^m where m is less than or equal to 3.
The paper utilizes the fact that the infimum of distances between a point and convex combinations of other points in a set is bounded below by 1/sqrt(2) in specific cases.

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by Krzysztof J.... at **arxiv.org** 10-07-2024

Deeper Inquiries

This paper makes several contributions to the understanding of Lipschitz extensions, which could potentially influence the development of numerical methods for optimal transport problems. Here's how:
Deeper understanding of optimality conditions: The paper explores conditions under which a given map allows for distance-preserving Lipschitz extensions. In the context of optimal transport, this relates to understanding when a transport plan is optimal. The characterization of such maps, as presented in Theorem 1.2, could potentially lead to new algorithms or improve existing ones by providing tighter optimality conditions.
Multi-dimensional Localization: The paper addresses the conjecture by Klartag regarding multi-dimensional localization in optimal transport. While the conjecture is disproved in general, the paper shows its validity under specific conditions related to the existence of distance-preserving extensions. This refined understanding of the limitations and applicability of localization techniques is crucial for developing efficient numerical methods, especially for high-dimensional problems.
New avenues for discretization: Numerical methods often rely on discretizing the underlying space. The paper's focus on the interplay between continuous and discrete settings, particularly in the context of isometric embeddings and leaves of Lipschitz maps, could inspire new discretization schemes. For instance, understanding the structure of leaves could lead to adaptive grids that better capture the geometry of the transport problem.
However, it's important to note that the paper's results are theoretical. Bridging the gap between these theoretical insights and practical numerical implementations will require further research.

Yes, exploring alternative characterizations of maps admitting distance-preserving Lipschitz extensions is a promising research direction. Here are some potential avenues:
Modulus of continuity: Instead of focusing solely on the Lipschitz constant (which bounds the global behavior), one could investigate the modulus of continuity. This function provides a more refined description of the local Lipschitz behavior. Characterizing maps with specific moduli of continuity that guarantee distance-preserving extensions could be insightful.
Differential properties: For maps defined on open sets, exploring connections between the existence of distance-preserving extensions and properties of their derivatives (or subdifferentials for non-smooth maps) could be fruitful. For instance, are there conditions on the Jacobian or the Hessian that guarantee such extensions?
Geometric measure theory: Tools from geometric measure theory, such as currents or varifolds, could provide a different perspective. These tools are well-suited for studying the geometry of sets and maps in a measure-theoretic setting, which might reveal new characterizations.
Curvature conditions: The paper mentions that the Kirszbraun theorem extends to spaces with bounded curvature in the sense of Alexandrov. Investigating how curvature conditions on the source and target spaces influence the existence and characterization of maps with distance-preserving extensions is an intriguing direction.
Exploring these alternative characterizations could lead to a deeper understanding of the geometry underlying Lipschitz extensions and potentially uncover connections to other areas of mathematics.

Let's denote the space of all Lipschitz maps between two metric spaces X and Y with Lipschitz constant at most L as LipL(X, Y). The subset E ⊂ LipL(X, Y) consisting of maps admitting distance-preserving extensions has a rich structure, but determining its precise geometry is a complex question. Here are some observations and potential directions:
Non-convexity: In general, E is not a convex set. Consider simple examples in Euclidean space: the average of two maps with distance-preserving extensions might not have the same property.
Closedness: Under suitable conditions on the source space X (e.g., X being compact), the set E is closed in LipL(X, Y) with respect to the uniform distance. This follows from the fact that pointwise limits of Lipschitz maps with a fixed Lipschitz constant are Lipschitz, and the distance-preserving property is also preserved under uniform convergence.
Relationship with isometries: The set E contains all isometries from X to Y. Understanding how E is "built around" the set of isometries is crucial. For instance, are there "neighborhoods" of isometries within E with a specific structure?
Dependence on the source and target spaces: The geometry of E heavily depends on the properties of X and Y. For example, if Y has negative curvature in a suitable sense, the set E might be significantly smaller compared to the case when Y is flat or positively curved.
Analyzing the geometric structure of E could involve tools from metric geometry, functional analysis, and topology. It's a challenging problem with potential connections to the geometry of Lipschitz function spaces and the properties of extensions of maps.

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