The Hausdorff dimension of the visible sets of connected compact sets

Details The Hausdorff dimension of the visible sets of connected compact sets The Hausdorff dimension of the visible sets of connected compact sets

2003-10-10

arxiv.org/abs/math/0310145v1

Mathematics

Classical Analysis and ODEs

Approximately 40 pages with 6 figures

Scientific paper

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.

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