Polygons inscribed in Jordan curves with prescribed edge ratios

Abstract

Let $J$ be a simple closed curve in $\mathbb R^{k}$ $(k\geq2)$ that is differentiable with non-zero derivative at a point $A_0\in J$. For a tuple of positive reals $a_1,\cdots,a_n$ $(n\geq3)$, each of which is less than the sum of the others, we show that there exists a polygon $Q_n$ inscribed in $J$ with sides of lengths proportional to $(a_1,\cdots,a_n)$. As a consequence, we prove the existence of triangle inscribed in $J$ similar to any given triangle.