Are lines much bigger than line segments?
Metric Geometry
2018-03-12 v3 Classical Analysis and ODEs
Abstract
We pose the following conjecture: (*) If A is the union of line segments in R^n, and B is the union of the corresponding full lines then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in R^n has Hausdorff dimension at least n-1 and (upper) Minkowski dimension n. We also prove that conjecture (*) holds if the Hausdorff dimension of B is at most 2, so in particular it holds in the plane.
Cite
@article{arxiv.1409.5992,
title = {Are lines much bigger than line segments?},
author = {Tamás Keleti},
journal= {arXiv preprint arXiv:1409.5992},
year = {2018}
}
Comments
minor corrections, "much" was added in the title