Discrete length-volume inequalities and lower volume bounds in metric spaces

Abstract

A theorem of W. Derrick ensures that the volume of any Riemannian cube \(([0,1]^n,g)\) is bounded below by the product of the distances between opposite codimension-1 faces. In this paper, we establish a discrete analog of Derrick’s inequality for weighted open covers of the cube \([0,1]^n\), which is motivated by a question about lower volume bounds in metric spaces. Our main theorem generalizes a previous result of the author in Kinneberg (J Differ Geom 100(2):349–388, 2015) which gave a combinatorial version of Derrick’s inequality and was used in the analysis of boundaries of hyperbolic groups. As an application, we answer a question of Y. Burago and V. Zalgaller about length-volume inequalities for pseudometrics on the unit cube.