Isoperimetric-type inequalities on constant curvature manifolds (Q2903475)

Let \((M,g)\), \((M^\prime, g^\prime)\) be two complete \(n\)-dimensional Riemannian manifolds equipped with their volume measures vol and vol\(^\prime\). For given two probability measures \(\mu=\varrho_0\text{vol}\) and \(\nu=\varrho_1\text{vol}^\prime\) on \(M\) and \(M^\prime\), respectively, a Borel map \(\Phi: M\to M^\prime\) is said to be a transport map if \(\Phi_\sharp\mu=\nu\). For a given function \(c: M\times M^\prime\to \mathbb R\), called a cost, a transport map \(\Phi\) is said to be an optimal transport map if it minimizes the total cost functional \(\mathcal{T}(\Phi)=\int_Mc(x, \Phi(x))d\mu(x).\) The authors, using the optimal transport method on Riemannian manifolds together with a suitably adopted Gromov's proof of the isoperimetric inequality in the Euclidean space, obtain an isoperimetric-type inequality on simply connected manifolds of constant sectional curvature.