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

TL;DR

This paper develops a discrete analog of Derrick's volume inequality for metric spaces, providing new lower volume bounds and answering open questions about length-volume inequalities in pseudometric spaces.

Contribution

It generalizes Derrick's inequality to discrete covers of cubes and applies this to establish lower volume bounds in metric spaces, extending previous combinatorial results.

Findings

Established a discrete Derrick-type inequality for weighted open covers.

Provided lower volume bounds for pseudometrics on the unit cube.

Answered a question about length-volume inequalities 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, 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.