Editing Isoperimetric dimension (section)

==Formal definition==

We say about a [[differentiable manifold]] ''M'' that it satisfies a ''d''-dimensional '''isoperimetric inequality''' if for any open set ''D'' in ''M'' with a smooth boundary one has

:<math>\operatorname{area}(\partial D)\geq C\operatorname{vol}(D)^{(d-1)/d}.</math>

The notations vol and area refer to the regular notions of volume and surface area on the manifold, or more precisely, if the manifold has ''n'' topological dimensions then vol refers to ''n''-dimensional volume and area refers to (''n''&nbsp;&minus;&nbsp;1)-dimensional volume. ''C'' here refers to some constant, which does not depend on ''D'' (it may depend on the manifold and on ''d'').

The '''isoperimetric dimension''' of ''M'' is the [[Infimum and supremum|supremum]] of all values of ''d'' such that ''M'' satisfies a ''d''-dimensional isoperimetric inequality.