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.

11 pages, 2 figures. arXiv admin note: substantial text overlap with arXiv:1402.6174

References in corpus (3)

• The discrete square peg problem
• On the number of inscribed squares of a simple closed curve in the plane
• The algebraic square peg problem