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

TL;DR

This paper generalizes Nielsen's theorem to higher dimensions, showing that certain triangles can be inscribed in Jordan curves in ^n, and explores conditions for inscribing equilateral triangles at specific points.

Contribution

It extends the inscribed triangle theorem from planar Jordan curves to ^n and investigates conditions for inscribing equilateral triangles at points on the curve.

Findings

Generalization of Nielsen's theorem to ^n for specific triangle sets

Identification of conditions for inscribing equilateral triangles at points

Extension of inscribed triangle concepts to higher-dimensional Jordan curves

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.