The stability of Almgren's isoperimetric inequality

We establish a quantitative isoperimetric inequality in higher codimension. In particular, we prove that for any closed (n-1)-dimensional manifold \Gamma in \R^{n+k} the following inequality $$D(\Gamma)\ge C d^2(\Gamma)$$ holds true. Here, D(\Gamma) stands for the isoperimetric gap of \Gamma, i.e. the deviation in measure of \Gamma from being a round sphere and d(\Gamma ) denotes a natural generalization of the Fraenkel asymmetry index of \Gamma to higher codimension.