Wei Hong HE(1), Jun LUO(1), Zu
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.