Haudorff测度与等径不等式

对于:Hausdorff维数为s>0的满足开集条件的自相似集E(?)Rn(n>1),我们引入等径不等式Hs|E(X)≤|X|s,以及使该不等式等号成立而直径大于0的极限集U(?)Rn.这里,Hs|E(·)是限制到集合E上的s维Hausdorff测度,而|X|指集合X在欧氏度量下的直径.当s=n时,n维球是唯一的极限集;当s∈(1,n)时,除去一些反面例子以外,我们对上述等径不等式的极限集的基本性质所知甚少.可以看出,这些不等式与Hs(E)的准确值的计算有密切联系.作为特例,我们将考虑Sierpinski垫片,指出计算这一典型自相似集的In2/In3维Hausdorff测度准确值的困难何在.由此可以大致推想,为什么除去平凡情形以外,至今还没有一个具体的满足开集条件而维数大于1的自相似集的:Hausdorff测度准确值被计算出来.

Hausdorff Measure and Isodiametric Inequalities

For self-similar sets E (?) n with the Open Set Condition and of Hausdorff dimension s > 0, we introduce the isodiametric inequality Hs|E(X) ≤|X|s and the corresponding extremal sets U (?) Rn with positive diameter such that Hs|E(U) = |U|s. Here Hs|E(·) is the s dimensional Hausdorff measure restricted to E, and |X| is the diameter of the set X in the standard Euclidean metric. If s = n, disks/balls are the unique extremal sets; if s ∈ (1,n), we have few ideas on properties of the extremal domains, but a few negative candidates. We can see that these isodiametric inequalities are related to the searching for exact value of Hs(E). Particularly, we take the Sierpinski gasket as an example, showing what the difficulty is or where it lies to find the the exact value of its In3/In2 dimensional Hausdorff measure. In some sense, this explains why, except for trivial examples, there are up to now no concrete self-similar sets with the Open Set Condition and of Hausdorff dimension larger than 1 such that the exact value of its Hausdorff measure has been calculated.