Polygons inscribed in Jordan curves with prescribed edge ratios (Q6581822)

This paper ultimately deals with the question: Does every simple closed curve have an inscribed polygon with any prescribed edge ratio? In particular, the authors extend the following result in [\textit{R. C. Penner}, Commun. Math. Phys. 113, 299--339 (1987; Zbl 0642.32012)]:\N\NGiven \(n \geq 3\) positive real numbers \(a_1,\dots,a_n\), each of which is less than the sum of the others, then there exists a convex polygon \(Q_n = A_0\cdots A_{n-1}\) inscribed in the unit circle such that the vector of distances \[(|A_0A_1|, |A_1A_2|, \dots , |A_{n-2}A_{n-1}|, |A_{n-1}A_0|)\] is proportional to \((a_1,\dots a_n)\). Moreover, \(Q_n\) is unique up to isometries.\N\NNamely, the authors are able to extend the existence part of Penner's result replacing ``unit circle'' by ``oriented simple closed curve in \(\mathbb{R}^k\) that is differentiable at some point \(A_0\)''.\N\NFurthermore, the authors explore the convex situation as well as the finiteness of the number of solutions to the problem.