Higher order Wirtinger inequalities, the isoperimetric deficit, and Minkowski-type inequalities
It is well-known that the classical isoperimetric inequality on the plane follows from the Wirtinger inequality, i.e. if a periodic function has mean zero, then its L2 norm is bounded by the L2 norm of its derivative. In the first part, by using simple Fourier analysis, we obtain sharp Wirtinger-type inequalities which involve the derivatives of arbitrary large order. This can be used to obtain upper and lower bounds for the isoperimetric deficit of a convex curve in terms of its curvature and its derivatives. In the second part, I will talk about the generalization of these inequalities to higher dimensions using spherical harmonics. As an application, we obtain some stability results of the Minkowski inequalities for convex hypersurfaces or for mixed volumes. Part of this talk is joint work with Hojoo Lee.