Discrete length-volume inequalities and lower volume bounds in metric spaces
arXiv:1410.5692
Abstract
A theorem of W. Derrick ensures that the volume of any Riemannian cube 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 , which is motivated by a question about lower volume bounds in metric spaces. Our main theorem generalizes a previous result of the author, 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.
22 pages; v2: Math. Z. online version 2015