A sharp quantitative isoperimetric inequality in higher codimension

Abstract

We establish a quantitative isoperimetric inequality in higher codimension. In particular, we prove that for any closed $(n-1)$-dimensional manifold $\Gamma$ in ${\mathbb{R}}^{n+k}$ the following inequality

holds true. Here, $\mathbf{D}(\Gamma )$ stands for the isoperimetric gap of $\Gamma$, i.e. the deviation in measure of $\Gamma$ from being a round sphere and $\mathbf{d}(\Gamma )$ denotes a natural generalization of the Fraenkel asymmetry index of $\Gamma$ to higher codimensions.