Non-uniformly continuous nearest point maps

2024/02/07 by Rubén Medina, Medina, Rubén, Andrés Quilis +1
Decision Sciences · Mathematics · #46B20 #46B80 #51F30 #54C15 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Fuzzy and Soft Set Theory #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2402.04747

openalex publication_date 2024/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We construct a Banach space satisfying that the nearest point map (also called proximity mapping or metric projection) onto any compact and convex subset is continuous but not uniformly continuous. The space we construct is locally uniformly convex, which ensures the continuity of all these nearest point maps. Moreover, we prove that every infinite-dimensional separable Banach space is arbitrarily close (in the Banach-Mazur distance) to one satisfying the above conditions.

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