Triangles on planar Jordan $C^1$-curves and differential topology

Abstract

We prove that a Jordan $\mathcal{C}^1$-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 $C1$. 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.