Rectangle Coincidences and Sweepouts
Metric Geometry
2018-11-28 v3
Abstract
We prove an integral formula for continuous paths of rectangles inscribed in a piecewise smooth loop. We then use this integral formula to show that (with a very mild genericity hypothesis) the number of rectangle coincidences, informally described as the number of inscribed rectangles minus the number of isometry classes of inscribed rectangles, grows linearly with the number of positively oriented extremal chords -- a.k.a. diameters -- of the polygon
Cite
@article{arxiv.1809.03070,
title = {Rectangle Coincidences and Sweepouts},
author = {Richard Evan Schwartz},
journal= {arXiv preprint arXiv:1809.03070},
year = {2018}
}
Comments
19 pages, traditional proof. This update corrects a glitch in which I left off the statement that the extremal chords need to be positively oriented, in a sense that is explained in the paper