A Trichotomy for Rectangles Inscribed in Jordan Loops
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.
Keywords
Cite
@article{arxiv.1804.00740,
title = {A Trichotomy for Rectangles Inscribed in Jordan Loops},
author = {Richard Evan Schwartz},
journal= {arXiv preprint arXiv:1804.00740},
year = {2019}
}
Comments
32 pages. This paper is a revision of the first version, inspired in part by comments from an anonymous referee. The current version of the paper will probably appear in Geometriae Dedicata