Illinois Journal of Mathematics

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

Yves Martinez-Maure

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We prove a Sturm-type comparison theorem by a geometric study of plane (multi)hedgehogs. This theorem implies that for every $2\pi$-periodic smooth real function $h$, the number of zeros of $h$ in $[0,2\pi[$ is not bigger than the number of zeros of $h+h^{\prime\prime}$ 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.

Article information

Illinois J. Math., Volume 52, Number 3 (2008), 981-993.

First available in Project Euclid: 1 October 2009

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Zentralblatt MATH identifier

Primary: 52A30: Variants of convex sets (star-shaped, (m, n)-convex, etc.) 53A04: Curves in Euclidean space


Martinez-Maure, Yves. A Sturm-type comparison theorem by a geometric study of plane multihedgehogs. Illinois J. Math. 52 (2008), no. 3, 981--993. doi:10.1215/ijm/1254403726.

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