Moduli of Continuity in Metric Models and Extension of Livability Indices

• Published in: Axioms Vol. 13; no. 3; p. 192
• Main Authors: Arnau, Roger, Calabuig, Jose M., González, Álvaro, Sánchez Pérez, Enrique A.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-03-2024
• Subjects: Composition; Continuity; Continuity (mathematics); extension; Functional equations; Functions; index; Lipschitz; livability; Mathematical analysis; metric; Metric space; Metric spaces; Spain
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms13030192

Index spaces serve as valuable metric models for studying properties relevant to various applications, such as social science or economics. These properties are represented by real Lipschitz functions that describe the degree of association with each element within the underlying metric space. After determining the index value within a given sample subset, the classic McShane and Whitney formulas allow a Lipschitz regression procedure to be performed to extend the index values over the entire metric space. To improve the adaptability of the metric model to specific scenarios, this paper introduces the concept of a composition metric, which involves composing a metric with an increasing, positive and subadditive function ϕ. The results presented here extend well-established results for Lipschitz indices on metric spaces to composition metrics. In addition, we establish the corresponding approximation properties that facilitate the use of this functional structure. To illustrate the power and simplicity of this mathematical framework, we provide a concrete application involving the modeling of livability indices in North American cities.