2003/10/10 by Toby C O'Neil, Toby C. O’Neil, O'Neil, Toby C
Computer Science · Engineering · Mathematics · #28A80 (Primary) 28A78 #31A15 (Secondary) #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Variational Analysis #math.CA #math.MG #msc:28A78 #msc:28A80 #msc:31A15
paper · pdf · doi:10.48550/arxiv.math/0310145
Approximately 40 pages with 6 figures
arxiv created 2003/10/10 · openalex publication_date 2003/10/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Kx=u∈ K : [x,u]∩ 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 Kx is strictly less than the Hausdorff dimension of K. In fact, for almost every x, dim(Kx)≤ 1/2+√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+√dim(K)-3/4, for s>0.