Curves of constant diameter and inscribed polygons
Metric Geometry
2012-02-14 v2
Abstract
A simple closed curve in the Euclidean plane is said to have property C_n(R) if at each point we can inscribe a unique regular -gon with edges length . C_2(R) is equivalent to having constant diameter. We show that smooth curves satisfying C_n(R) other than the circle do exist for all n, and that the circle is the only regular curve satisfying C_2(R) and C_4(R') where . In an addendum, we show that the last assertion holds for any R and R'. The proofs use only elementary differential calculus and geometry.
Cite
@article{arxiv.math/0504300,
title = {Curves of constant diameter and inscribed polygons},
author = {Mathieu Baillif},
journal= {arXiv preprint arXiv:math/0504300},
year = {2012}
}
Comments
6 pages, 10 figures. Aimed at an undergraduate audiance. V2 includes the published version, with many suggestions from the referee, and a short addendum with some improvements of the results