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A sharp quantitative isoperimetric inequality in higher codimension. (English) Zbl 1320.49030

Summary: 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 \[ \mathbf{D}(\Gamma)\geq C \mathbf{d}^2(\Gamma) \] 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.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
52A40 Inequalities and extremum problems involving convexity in convex geometry
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