A Trichotomy for Rectangles Inscribed in Jordan Loops

TL;DR

This paper establishes a trichotomy for rectangles inscribed in Jordan loops, showing that such rectangles form connected sets with specific geometric properties, depending on the loop's structure.

Contribution

It introduces a novel trichotomy classification for inscribed rectangles in Jordan loops, revealing three distinct geometric configurations.

Findings

Existence of a connected set of rectangles with large area including a square

Presence of rectangles with all aspect ratios covering almost all points of g

Availability of small-diameter rectangles covering almost all points of g

Abstract

Let g be an arbitrary Jordan loop and let G denote the space of rectangles R which are inscribed in g in such a way that the cyclic order of the vertices of R is the same whether it is induced by R or by g. We prove that G contains a connected set S satisfying one of three properties: 1. S consists of rectangles of uniformly large area, including a square, and every point of g is the vertex of a rectangle in S. 2. S consists of rectangles having all possible aspect ratios, and all but at most 4 points of g are vertices of rectangles in S. 3. S contains rectangles of every sufficiently small diameter, and all but at most 2 points of g are vertices of rectangles in S.