中文摘要：

本文证明对于满足强分离条件的自相似集E,存在一个闭凸集达到它的最大密度.即存在一个闭凸集V E0（｜V｜〉0）,使得sup{μ（U）/｜U｜^s：U E0}=μ（V）/｜V｜s,其中U为闭集,E0表示自相似集E的闭凸壳,μ表示E上的唯一自相似概率测度.作为应用,我们给出命题“满足强分离开集条件的自相似集具有最优几乎处处覆盖”的一个新证明.

英文摘要：

We prove that there exists a closed convex set attaining the maximum density for the self-similar set E satisfying the strong separation condition. That is, there exists a closed convex set V C Eo, with ｜V｜ 〉 0, such that sup{μ（U）/｜U｜^s：U E0 is closed} =μ（U）/｜U｜^s, where E0 denotes the closed convex hull of the self-similar set E and μ denotes the unique self-similar probability measure on E. As a result, we give a new proof of the proposition that any self-similax set with the strong separation condition （SSC） admits an optional almost covering.