2021/03/25 by Cantarella, Jason, Denne, Elizabeth, McCleary, John
#51M04 #57Q65 #58A20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Primary 53A04 #Secondary 55R80
paper · doi:10.48550/arxiv.2103.13848
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 C1-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.