Best constants for the isoperimetric inequality in quantitative form

TL;DR

This paper develops a method to determine the optimal constants in the quantitative isoperimetric inequality, explicitly computing them in two dimensions using convex sets called ovals, thus extending a previous conjecture.

Contribution

It introduces a new approach combining existence, regularity, and symmetrization techniques to find best constants in the quantitative isoperimetric inequality, especially in 2D.

Findings

Explicit computation of constants for n=2 using ovals

Extension of Hall's conjecture in the plane

Method applicable to higher order terms in inequalities

Abstract

We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in $R^n$. Combining these results with a refinement of the selection principle introduced by the authors in arXiv:0911.0786, we describe a method suitable for the determination of the best constants in the quantitative isoperimetric inequality with higher order terms. Then, applying Bonnesen's annular symmetrization in a very elementary way, we show that, for $n=2$, the above-mentioned constants can be explicitly computed through a one-parameter family of convex sets known as ovals. This proves a further extension of a conjecture posed by Hall in J. Reine Angew. Math. 428 (1992).