2025/01/04 by Donghan Kim, Kim, Donghan
Mathematics · #Advanced Differential Equations and Dynamical Systems #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #General Topology (math.GN) #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2501.02259
openalex publication_date 2025/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we address the longstanding question of whether expansive homeomorphisms can exist within convex bodies in Euclidean spaces. Utilizing fundamental tools from topology, including the Borsuk-Ulam theorem and Brouwer's fixed-point theorem, we establish the nonexistence of such mappings. Through an inductive approach based on dimension and the extension of boundary homeomorphisms, we demonstrate that expansive homeomorphisms are incompatible with the compact and convex structure of these bodies. This work highlights the interplay between topological principles and metric geometry, offering new insights into the constraints imposed by convexity.