Closed polylines inscribed in a closed space curve

Abstract

Let n be an odd positive integer. It is proved that if n + 2 is a power of a prime number and C is a regular closed non-self-intersecting curve in $ {\mathbb{R}^n} $ ,then C contains vertices of an equilateral (n + 2)-link polyline with n + 1 vertices lying in a hyperplane. It is also proved that if C is a rectifiable closed curve in $ {\mathbb{R}^n} $ ,then C contains n + 1 points that lie in a hyperplane and divide C into parts one of which is twice as long as each of the others. Bibliography: 6 titles.