Maximum density for the Sierpinski carpet
We prove that there exists a closed convex set obtaining the maximum density for the Sierpinski carpet S S . That is, there exists a closed convex set V?E0 V ? E 0 , with |V|>0 | V | > 0 , such that sup { ? ( U ) | U | s : U ? E 0 , isclosed } = ? ( V ) | V | s , where E0 E 0 is defined in the introduction and ? ? denotes the unique self-similar probability measure on S S . We give a reasonable description about the shape of V V .
density sierpinski carpet hausdorff metric