Abstract
In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set Ω , different from a ball, which minimizes the ratio δ ( Ω ) / λ 2 ( Ω ), where δ is the isoperimetric deficit and λ the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.
Highlights
The last few years have seen several remarkable breakthroughs in the study of quantitative isoperimetric inequalities
Its isoperimetric deficit is defined as δ(Ω)
The isoperimetric inequality guarantees that δ(Ω) is positive and null only if Ω is a ball
Summary
The last few years have seen several remarkable breakthroughs in the study of quantitative isoperimetric inequalities. Isoperimetric inequality, quantitative isoperimetric inequality, isoperimetric deficit, Fraenkel asymmetry, rearrangement, shape derivative, optimality conditions This existence result provides a new proof of the quantitative isoperimetric inequality in the plane. Let C be the class of planar convex sets; inf F(Ω) = 0.405585 , Ω∈C and the minimum is attained at a particular “stadium” This optimal stadium will be useful for us in excluding possible minimizing sequences converging to the ball. By using an iterative selection principle and by applying Bonnesen’s annular symmetrization, they showed that a minimizing sequence for the above infimum is made up of ovals, that is, C1 convex sets, with two orthogonal axes of symmetry, whose boundary is the union of two congruent arcs of circle They proved this result using properties of this family of sets established in [3] and [1]. We explain what remains to show that the mask is an optimal set for F
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