The Hausdorff dimension of the visible sets of connected compact sets
| dc.creator | O'Neil, Toby C | |
| dc.date | 2003-10-10 | |
| dc.date.accessioned | 2026-07-07T05:01:47Z | |
| dc.date.available | 2026-07-07T05:01:47Z | |
| dc.description | For a compact subset K of the plane and a point x, we define the visible part of K from x to be the set K_x={u\in K : [x,u]\cap K={u}}. (Here [x,u] denotes the closed line segment joining x to u.) In this paper, we use energies to show that if K is a compact connected set of Hausdorff dimension larger than one, then for (Lebesgue) almost every point x in the plane, the Hausdorff dimension of K_x is strictly less than the Hausdorff dimension of K. In fact, for almost every x, dim(K_x)\leq {1/2}+\sqrt{dim(K)-{3/4}}. We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension larger than s+{1/2}+\sqrt{dim(K)-{3/4}}, for s>0. | |
| dc.description | Approximately 40 pages with 6 figures | |
| dc.identifier | https://arxiv.org/abs/math/0310145 | |
| dc.identifier | http://arxiv.org/abs/math/0310145 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/68805 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Metric Geometry | |
| dc.subject | 28A80 (Primary) 28A78, 31A15 (Secondary) | |
| dc.title | The Hausdorff dimension of the visible sets of connected compact sets | |
| dc.type | text |