Maximum density for the Sierpinski carpet

We prove that there exists a closed convex set obtaining the maximum density for the Sierpinski carpet SS. That is, there exists a closed convex set V⊂E0V⊂E0, with |V|>0|V|>0, such that sup{μ(U)|U|s:U⊂E0,is closed}=μ(V)|V|s, where E0E0 is defined in the introduction and μμ denotes the unique self-similar probability measure on SS. We give a reasonable description about the shape of VV.