On a closed Riemannian manifold, the Euclidean quantitative isoperimetric inequality is false in general, holds for generic metrics with the sharp exponent, and holds for real analytic metrics with a modified sharp exponent.
Carlen and Alessio Figalli
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The Riemannian Quantitative Isoperimetric Inequality
On a closed Riemannian manifold, the Euclidean quantitative isoperimetric inequality is false in general, holds for generic metrics with the sharp exponent, and holds for real analytic metrics with a modified sharp exponent.