A Sturm-type comparison theorem by a geometric study of plane multihedgehogs (Q733367)

The author proves a comparison theorem for real periodic functions. Let \(h\) be a smooth real \(2N\pi\)-periodic function, let \(n_h\) be the number of zeros of \(h\) in \([0,2N\pi]\) and let \(S\) be the corresponding number of zeros of \(h+h''\). Then, \(n_h\leq S+4N-2\) holds. The proof uses a geometric interpretation of periodic functions. A smooth \(2\pi\)-periodic function is a difference of support functions and corresponds therefore to the difference of planar convex bodies. The latter is called a hedgehog, it can be visualized through the envelop map as a curve in \({\mathbb R}^2\). Here, a corresponding generalization, a multihedgehog is used. The theorem then implies an inequality between the number of oriented support lines and the number of singularities of the multihedgehog.