Are lines much bigger than line segments?

摘要

We pose the following conjecture: (*) If A is the union of line segments in R^n, and B is the union of thecorresponding full lines then the Hausdorff dimensions of A and B agree. We prove that this conjecture would imply that every Besicovitch set (compactset that contains line segments in every direction) in R^n has Hausdorffdimension at least n-1 and (upper) Minkowski dimension n. We also prove thatconjecture (*) holds if the Hausdorff dimension of B is at most 2, so inparticular it holds in the plane.