Triangles on planar Jordan $C^1$-curves

Abstract

We prove that a Jordan $\calc^1$-curve in the plane contains any non-flat triangle up to translation and homothety with positive ratio. This is false if the curve is not $C^1$. The proof uses a bit configuration spaces, differential and algebraic topology as well as the Schoenflies theorem. A partial generalization holds true in higher dimensions.

13 pages, 2 fugures. A new section (Section 5) has been added containing a partial generalization of the main theorem in higher dimensions