An isoperimetric inequality in \({\mathbb{R}}^ 3\) (Q1100725)

The classical isoperimetric inequality for domains T in \({\mathbb{R}}^ 4\) is generalized to the case where the closed hypersurface \(S=\partial T\) is given as the image \(\Phi ({\mathbb{R}}^ 3)\) for some \(\Phi:{\mathbb{R}}^ 3\to {\mathbb{R}}^ 4\). These functions \(\Phi\) have to satisfy certain integrability conditions and an approximation property, and the isoperimetric inequality becomes an integral inequality.