Triangles on planar Jordan $C^1$-curves

TL;DR

The paper proves that any non-flat triangle can be inscribed in a planar Jordan $C^1$-curve through translation and scaling, using topology and geometric arguments, with partial higher-dimensional extensions.

Contribution

It establishes the inscribability of all non-flat triangles in $C^1$ Jordan curves, highlighting the importance of smoothness for this property.

Findings

Any non-flat triangle can be inscribed in a $C^1$ Jordan curve.

The property fails for curves that are not $C^1$.

Partial higher-dimensional generalizations are possible.

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.