Tout chemin générique de hérissons réalisant un retournement de la sphère dans $\mathbb{R}^{3}$ comprend un hérisson porteur de queues d'aronde positives

Abstract

Hedgehogs are (possibly singular and self-intersecting) hypersurfaces that describe Minkowski differences of convex bodies in $\mathbb{R}^{n+1}$. They are the natural geometrical objects when one seeks to extend parts of the Brunn-Minkowski theory to a vector space which contains convex bodies. In this paper, we prove that in every generic path of hedgehogs performing the eversion of the sphere in $\mathbb{R}^{3}$, there exists a hedgehog that has positive swallowtails. This study was motivated by an open problem raised in 1985 by Langevin, Levitt, and Rosenberg.