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Let $E$ be a self-similar set satisfying the open set condition. Zhou and Feng posed an open problem in 2004 as follows: let $x ∈ E,$ under what conditions is there a set $U_x$ containing $x$ with $|U_x| > 0$ such that $\overline{D}^s_C(E,x) = \frac{\mathcal{H}^s(E \cap U_x)}{|U_x|^s}?$ The aim of this paper is to present a solution of this problem. Under the assumption that there exists a nonempty convex open set $V$ containing $E$ and satisfying the requirement of the open set condition, it is proved that if $x ∈ E$ and the upper convex density of $E$ at $x$ equals 1, then there exists a convex set $U_x$ containing $x$ with $|U_x| > 0$ such that $\overline{D}^s_C(E,x) =\frac{\mathcal{H}^s (E∩U_x)}{|U_x|^s}.$ Finally, as an application of this result, an equivalent condition for $E_0 = E$ is given, where $E_0 =\{x∈E|\overline{D}^s_C(E,x)=1\}.$
}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v57n4.24.07}, url = {http://global-sci.org/intro/article_detail/jms/23714.html} }Let $E$ be a self-similar set satisfying the open set condition. Zhou and Feng posed an open problem in 2004 as follows: let $x ∈ E,$ under what conditions is there a set $U_x$ containing $x$ with $|U_x| > 0$ such that $\overline{D}^s_C(E,x) = \frac{\mathcal{H}^s(E \cap U_x)}{|U_x|^s}?$ The aim of this paper is to present a solution of this problem. Under the assumption that there exists a nonempty convex open set $V$ containing $E$ and satisfying the requirement of the open set condition, it is proved that if $x ∈ E$ and the upper convex density of $E$ at $x$ equals 1, then there exists a convex set $U_x$ containing $x$ with $|U_x| > 0$ such that $\overline{D}^s_C(E,x) =\frac{\mathcal{H}^s (E∩U_x)}{|U_x|^s}.$ Finally, as an application of this result, an equivalent condition for $E_0 = E$ is given, where $E_0 =\{x∈E|\overline{D}^s_C(E,x)=1\}.$