A unified approach to higher order discrete and smooth isoperimetric inequalities

TL;DR

This paper introduces a unified Fourier analysis-based method to derive high-order isoperimetric inequalities for both discrete polygons and smooth curves, providing bounds and relations involving curvature and geometric measures.

Contribution

It presents a novel unified approach to high-order isoperimetric inequalities for discrete and smooth cases using Fourier analysis and linear operators.

Findings

Derived high-order discrete polygonal isoperimetric inequalities.

Established bounds for isoperimetric deficit in smooth curves.

Formulated higher order Chernoff type inequalities.

Abstract

We provide a simple unified approach to obtain (i) Discrete polygonal isoperimetric type inequalities of arbitrary high order. (ii) Arbitrary high order isoperimetric type inequalities for smooth curves, where both upper and lower bounds for the isoperimetric deficit $L^2-4\pi F$ are obtained. (iii) Higher order Chernoff type inequalities involving a generalized width function and higher order locus of curvature centers. The method we use is to obtain higher order discrete or smooth Wirtinger inequalities via discrete or smooth Fourier analysis, by looking at a family of linear operators. The key is to find the right candidate for the linear operators, and to translate the analytic inequalities into geometric ones.