# Isoperimetric inequality on a metric measure space and Lipschitz order with an additive error

## TL;DR

This paper extends Gromov's Lipschitz order to include an additive error, enabling the derivation of isoperimetric inequalities on non-discrete spaces from discrete approximations, specifically applied to the $l^1$-cube.

## Contribution

It introduces a relaxed Lipschitz order with additive error and applies it to establish isoperimetric inequalities on non-discrete metric measure spaces.

## Key findings

- Established an isoperimetric inequality on the non-discrete $l^1$-cube.
- Extended Gromov's Lipschitz order to include additive errors.
- Connected discrete and non-discrete isoperimetric inequalities through limits.

## Abstract

M. Gromov introduced the Lipschitz order relation on the set of metric measure spaces and developed a rich theory. In particular, he claimed that an isoperimetric inequality on a non-discrete space is represented by using the Lipschitz order. We relax the definition of the Lipschitz order allowing an additive error to relate with an isoperimetric inequality on a discrete space. As an application, we obtain an isoperimetric inequality on the non-discrete $n$-dimensional $l^1$-cube by taking the limits of an isoperimetric inequality of the discrete $l^1$-cubes.