  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.
