2014/10/21 by Kyle Kinneberg, Kinneberg, Kyle
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Primary: 52C17 #Secondary: 30L99
paper · pdf · doi:10.48550/arxiv.1410.5692
openalex publication_date 2014/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A theorem of W. Derrick ensures that the volume of any Riemannian cube\n([0,1]n,g) is bounded below by the product of the distances between opposite\ncodimension-1 faces. In this paper, we establish a discrete analog of Derrick's\ninequality for weighted open covers of the cube [0,1]n, which is motivated\nby a question about lower volume bounds in metric spaces. Our main theorem\ngeneralizes a previous result of the author, which gave a combinatorial version\nof Derrick's inequality and was used in the analysis of boundaries of\nhyperbolic groups. As an application, we answer a question of Y. Burago and V.\nZalgaller about length-volume inequalities for pseudometrics on the unit cube.\n