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

Abstract

We prove a Sturm-type comparison theorem by a geometric study of plane (multi)hedgehogs. This theorem implies that for every 2π-periodic smooth real function h, the number of zeros of h in [0, 2π[ is not bigger than the number of zeros of h+h^′′ plus 2. In terms of N-hedgehogs, it can be interpreted as a comparison theorem between number of singularities and maximal number of support lines through a point. The rest of the paper is devoted to a series of geometric consequences.