In this work, for the self similar sets satisfying the open set condition, we obtain an interesting character that the upper convex density at every similarly contractive fixed point equalling 1 implies that the upper convex densities at all the points are equal to 1. By using this result, we sufficiently answer the open problem 6 posed by Z. Zhou and L. Feng in [Twelve open problems on the exact value of the Hausdor measure and on topological entropy: A brief survey of recent results, Nonlinearity, 17(2004) 493–502.].
"Upper convex density and similarly contractive fixed points." Tbilisi Math. J. 5 (1) 39 - 49, 2012. https://doi.org/10.32513/tbilisi/1528768888