Haudorff测度与等径不等式  

Hausdorff Measure and Isodiametric Inequalities

作　　者：何伟弘[1] 罗俊[1] 周作领[2] 

机构地区：[1]中山大学数学与计算科学学院,广州510275 [2]中山大学岭南学院,广州510275

出　　处：《数学学报（中文版）》2005年第5期939-946,共8页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金(10041005)广东省自然科学基金(011221)中山大学高等研究中心基金(01M2;05M14)

摘　　要：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.For self-similar sets E belong to R^n with the Open Set Condition and of Hausdorff dimension s 〉 0, we introduce the isodiametric inequality H^s│E（X）≤│X│^s and the corresponding extremal sets U belong to R^n with positive diameter such that H^s│E（U）=│U│^s. Here H^s│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 H^s（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.

关 键 词：Hausodrff测度 等径不等式 部分估计原理 

分 类 号：O174.12[理学—数学]