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 Hausdorff measure and on topological entropy: A brief survey of recent results, Nonlinearity 17(2004), 493-502].

Tbilisi Mathematical Journal, Vol. 5(1) (2012), pp. 39-49