Are lines much bigger than line segments?
  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. 
Tam\'as Keleti
math.MG math.CA