中文摘要：

对于Hausdorff维数为s〉0的满足开集条件的自相似集E属于R^n（n〉1），我们引入等径不等式H^s│IE（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属于R^n．这里，H^s│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H^s（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约黑维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．

英文摘要：

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.