A Note On The Isoperimetric Inequality And Its Stability

TL;DR

This paper explores isoperimetric inequalities for convex curves in the plane, deriving new parametric inequalities and analyzing their stability, showing near equality indicates the curve is close to a circle.

Contribution

It introduces a family of parametric inequalities involving key geometric functionals of convex curves with a Fourier series proof and studies their stability properties.

Findings

Derived new parametric inequalities for convex curves.

Established stability results indicating near equality implies near circularity.

Connected inequalities to geometric Bonnesen-type inequalities.

Abstract

In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve with a simple Fourier series proof: length, area of the region included by the curve, area of the domain enclosed by the locus of curvature centers and integral of the radius of curvature. By using our isoperimetric-type inequalities, we also obtain some new geometric Bonnesen-type inequalities. Furthermore we investigate stability properties of such inequalities (near equality implies curve nearly circular).