A unified approach to higher order discrete and smooth isoperimetric inequalities

We present a unified approach to derive sharp isoperimetric-type inequalities of arbitrary high order. In particular, we obtain (i) sharp high-order discrete polygonal isoperimetric-type inequalities, (ii) sharp high-order isoperimetric-type inequalities for smooth curves with both upper and lower bounds for the isoperimetric deficit, and (iii) sharp higher order Chernoff-type inequalities involving a generalized width function and higher order locus of curvature centers. Our approach involves obtaining higher order discrete or smooth Wirtinger inequalities via Fourier analysis, by examining a family of linear operators. The key to our approach is identifying the appropriate linear operator and translating the analytic inequalities into geometric ones.