Sets that contain their circle centers
| dc.creator | Martin, Greg | |
| dc.date | 2007-03-29 | |
| dc.date.accessioned | 2026-07-07T07:55:02Z | |
| dc.date.available | 2026-07-07T07:55:02Z | |
| dc.description | Say that a subset S of the plane is a "circle-center set" if S is not a subset of a line, and whenever we choose three noncollinear points from S, the center of the unique circle through those three points is also an element of S. A problem appearing on the Macalester College Problem of the Week website was to prove that a finite set of points in the plane, no three lying on a common line, cannot be a circle-center set. Various solutions to this problem that did not use the full strength of the hypotheses appeared, and the conjecture was subsequently made that every circle-center set is unbounded. In this article, we prove a stronger assertion, namely that every circle-center set is dense in the plane, or equivalently that the only closed circle-center set is the entire plane. Along the way we show connections between our geometrical method of proof and number theory, real analysis, and topology. | |
| dc.description | 12 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/math/0703860 | |
| dc.identifier | http://arxiv.org/abs/math/0703860 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/126835 | |
| dc.subject | Metric Geometry | |
| dc.subject | 51M04 | |
| dc.title | Sets that contain their circle centers | |
| dc.type | text |