Inscribed triangles of Jordan curves in $\mathbb{R}^{n}$

Abstract

Nielsen's theorem states that any triangle can be inscribed in a planar Jordan curve. We prove a generalisation of this theorem, extending to any Jordan curve $J$ embedded in $\mathbb{R}^{n}$, for a restricted set of triangles. We then conclude by investigating a condition under which a given point of $J$ inscribes an equilateral triangle in particular.