A sharp quantitative isoperimetric inequality in higher codimension

  • Verena Bögelein

    Universität Salzburg, Austria
  • Frank Duzaar

    Universität Erlangen-Nünberg, Germany
  • Nicola Fusco

    Università degli Studi di Napoli Federico II, Italy

Abstract

We establish a quantitative isoperimetric inequality in higher codimension. In particular, we prove that for any closed (n1)(n-1)-dimensional manifold Γ\Gamma in Rn+k\mathbb R^{n+k} the following inequality

D(Γ)Cd2(Γ)\mathbf D(\Gamma)\ge C \mathbf d^2(\Gamma)

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

Cite this article

Verena Bögelein, Frank Duzaar, Nicola Fusco, A sharp quantitative isoperimetric inequality in higher codimension. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 26 (2015), no. 3, pp. 309–362

DOI 10.4171/RLM/709