Higher order Wirtinger-type inequalities and sharp bounds for the isoperimetric deficit

TL;DR

This paper develops high-order Wirtinger-type inequalities using Fourier analysis and applies them to establish sharp bounds for the isoperimetric deficit of closed curves in the Euclidean plane.

Contribution

It introduces a general method to derive high-order inequalities and provides new sharp bounds for geometric quantities related to closed plane curves.

Findings

Derived high-order Wirtinger-type inequalities using Fourier analysis

Established sharp lower and upper bounds for the isoperimetric deficit

Extended classical inequalities to higher orders in geometric analysis

Abstract

Using Fourier analysis, we derive Wirtinger-type inequalities of arbitrary high order. As applications, we prove various sharp geometric inequalities for closed curves on the Euclidean plane. In particular, we obtain both sharp lower and upper bounds for the isoperimetric deficit.