An overtwisted convex hypersurface in higher dimensions

We show that the germ of the contact structure surrounding a certain kind of convex hypersurface is overtwisted.We then find such hypersurfaces close to any plastikstufe with toric core, thereby proving that the existence of a plastikstufe with toric core implies overtwistedness.All proofs in this article are explicit, and we hope that the methods used here might hint at a deeper understanding of the size of neighborhoods in contact manifolds.In the appendix we reprove in a concise way that the Legendrian unknot is loose if the ambient manifold contains a large enough neighborhood of a 2-dimensional overtwisted disk.Additionally we prove the folklore result that the singular distribution induced on a hypersurface of a contact manifold .M; / determines the germ of the contact structure around .