Triangles on planar Jordan C1C^1-curves and differential topology

  • Jean-Claude Hausmann

    Université de Genève, Switzerland
Triangles on planar Jordan $C^1$-curves and differential topology cover

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We prove that a Jordan C1\mathcal{C}^1-curve in the plane contains the vertices of any non-flat triangle, up to translation and homothety with positive ratio. This is false if the curve is not C1C1. The proof makes use of configuration spaces, differential and algebraic topology as well as the smooth Schoenflies theorem. A partial generalization holds true in higher dimensions.