Isoperimetric inequality on a metric measure space and Lipschitz order with an additive error (Q2074488)

Isoperimetric inequalities are interesting geometric inequalities, and have been studied in various cases, e.g., Riemannian manifolds, and metric measure spaces with lower Ricci curvature bounds etc. For example, the classical Lévy isoperimetric inequality asserts that if \(\Omega\) is a domain in the standard sphere \(S^n\), and \(B_\Omega\) is the metric ball with \(m(\Omega)=m(B_\Omega)\) where \(m\) denotes the Riemannian measure, then their \(r\)-neighbourhoods satisfy \( m(U_r(\Omega))\geq m(U_r(B_\Omega)),\) for any \(r>0\). This paper proves isoperimetric inequalities on the \(n\)-dimensional cubes and the \(n\)-torus equipped with the \(l^1\)-metric. One of the main theorems shows that if \(\Omega\) is a domain in the \(l^1\)-torus \(T^n\), and \(B_\Omega\subset T^n\) is the metric ball with \(m(\Omega)=m(B_\Omega)\), then \[ m(U_r(\Omega))\geq m(U_r(B_\Omega)),\] for any \(r>0\). These isoperimetric inequalities are proved by taking the limits of isoperimetric inequalities of discrete \(l^1\)-cubes and \(l^1\)-tori under the so-called concentrate topology. The main technical result is the stability of isoperimetric inequalities with respect to the concentrate topology. The method builds on generalising the notion of the Lipschitz order introduced by Gromov.