Square-like quadrilaterals inscribed in embedded space curves

Abstract

The square-peg problem asks if every Jordan curve in the plane has four points which are the vertices of a square. The problem is open for continuous Jordan curves, but it has been resolved for various regularity classes of curves between continuous and C 1 -smooth Jordan curves.Here, in a generalization of the square-peg problem, we consider embedded curves in space, and ask if they have inscribed quadrilaterals with equal sides and equal diagonals. We call these quadrilaterals "square-like". We give a regularity class (finite total curvature without cusps) in which we can prove that every embedded curve has an inscribed square-like quadrilateral. The key idea is to use local data to show that short enough arcs have small curvature, thus ruling out small squares. This allows us to successfully use a limiting argument on approximating curves.Conjecture 1. There is a square-like quadrilateral inscribed in any embedding of S 1 in R n .

Families of similar simplices inscribed in most smoothly embedded spheres

“…This question was posed by Toeplitz in 1911 [44], and progress on this problem has chiefly been an extension of the regularity class of simple closed curves for which the square can be found. The interested reader can find numerous articles [9,26,32,36,42] summarising the problem and describing the classes of curves for which the square-peg problem has been proved. There have also been many papers [1,3,15,22,23,28,33,39,45] examining quadrilaterals and polygons inscribed in curves and, more recently, making progress towards solving the rectangular-peg problem (finding rectangles of any aspect ratio inscribed in Jordan curves).…”

Families of similar simplices inscribed in most smoothly embedded spheres

“…This question was posed by O. Toeplitz in 1911 [41], and progress on this problem has chiefly been extension of the regularity class of simple closed curves for which the square can be found. The interested reader can find numerous articles [10,25,31,35,40] summarizing the problem, and describing the classes of curves for which the square-peg problem has been proved. There have also been many papers [1,3,14,21,22,27,32,37,42] examining quadrilaterals and polygons inscribed in curves and, more recently, making progress towards solving the rectangular-peg problem (finding rectangles of any aspect ratio inscribed in Jordan curves).…”