直线是否比线段大得多？
Are lines much bigger than line segments?
摘要：
我们提出以下猜想：（*）如果A是R^n中的线段的并，B是相应的完整直线的并，则A和B的豪斯多夫维数一致。我们证明了这个猜想意味着R^n中的每个Besicovitch集（在每个方向上包含线段的紧集）至少有n-1的豪斯多夫维数和（上）Minkowski维数n.我们还证明了猜想（*）在B的豪斯多夫维数至多为2时成立，特别是在平面上成立.
英文摘要：
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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