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arXiv · 1409.5992

Are lines much bigger than line segments?

Tamás Keleti
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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.

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Tamás Keleti. 2014-09-21. Are lines much bigger than line segments?. https://doi.org/10.1090/proc/12978

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