2020/09/09 by Alexey Balitskiy, Balitskiy, Alexey, Aleksandr Berdnikov +1
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2009.04558
openalex publication_date 2020/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We discuss various questions of the following kind: for a continuous map X → Y from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The d-width measures how well a space can be approximated by a d-dimensional complex. The results of this paper include the following. 1) Any piecewise linear map f: [0,1]m+2 → Ym from the unit euclidean (m+2)-cube to an m-polyhedron must have a fiber of 1-width at least (1)/(2βm +m2 + m + 1), where β= supy rk H1(f-1(y)) measures the topological complexity of the map. 2) There exists a piecewise smooth map X3m+1 → ℝm, with X a riemannian (3m+1)-manifold of large 3m-width, and with all fibers being topological (2m+1)-balls of arbitrarily small (m+1)-width.