Square-like quadrilaterals inscribed in embedded space curves

TL;DR

This paper extends the classical square-peg problem to space curves, proving that under certain regularity conditions, every embedded space curve contains a quadrilateral with equal sides and diagonals, called square-like.

Contribution

It introduces a new regularity class for space curves and proves the existence of inscribed square-like quadrilaterals within this class, generalizing the planar problem.

Findings

Every embedded curve with finite total curvature has an inscribed square-like quadrilateral.

Short arcs with small curvature prevent the existence of small inscribed squares.

Limiting arguments on approximating curves establish the main existence result.

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.