A unified approach to higher order discrete and smooth isoperimetric 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.