Best constants for the isoperimetric inequality in quantitative form

  • Marco Cicalese

    Università degli Studi di Napoli Federico II, Italy
  • Gian Paolo Leonardi

    Università di Modena e Reggio Emilia, Italy


We prove some results in the context of isoperimetric inequalities with quantitative terms. In the 22-dimensional case, our main contribution is a method for determining the optimal coefficients c1,,cmc_{1},\dots,c_{m} in the inequality δP(E)k=1mckα(E)k+o(α(E)m)\delta P(E) \geq \sum_{k=1}^{m}c_{k}\alpha (E)^{k} + o(\alpha (E)^{m}), valid for each Borel set EE with positive and finite area, with δP(E)\delta P(E) and α(E)\alpha (E) being, respectively, the \textit{isoperimetric deficit} and the \textit{Fraenkel asymmetry} of EE. In nn dimensions, besides proving existence and regularity properties of minimizers for a wide class of \textit{quantitative isoperimetric quotients} including the lower semicontinuous extension of δP(E)α(E)2\frac{\delta P(E)}{\alpha (E)^{2}}, we describe the general technique upon which our 22-dimensional result is based. This technique, called Iterative Selection Principle, extends the one introduced in [12].

Cite this article

Marco Cicalese, Gian Paolo Leonardi, Best constants for the isoperimetric inequality in quantitative form. J. Eur. Math. Soc. 15 (2013), no. 3, pp. 1101–1129

DOI 10.4171/JEMS/387