2015/03/31 by Peter S. Landweber, Emanuel A. Lazar, Neel Patel
Mathematics · #Bounded function #Combinatorics #Composite material #Fiber #Geometric and Algebraic Topology #Materials science #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Physics #Point processes and geometric inequalities #math.AT #math.CA #math.MG
paper · pdf · doi:10.4169/amer.math.monthly.123.4.392
published as Amer. Math. Monthly, 123:4 (2016) · 6 pages, 2 figures
arxiv created 2015/10/30 · openalex publication_date 2016/01/01 · arxiv updated 2016/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We present a surprisingly short proof that for any continuous map f: ℝn → ℝ;m, if n > m, then there exists no bound on the diameter of fibers of f. Moreover, we show that when m = 1, the union of small fibers of f is bounded; when m > 1, the union of small fibers need not be bounded. Applications to data analysis are considered.