For a self-similar set $E$ satisfying the open set condition, upper convex density is an important concept for the computation of its Hausdorff measure, and it is well known that the set of relative interior points with upper convex density 1 has a full Hausdorff measure. But whether the upper convex densities of $E$ at all the relative interior points are equal to 1? In other words, whether there exists a relative interior point of $E$ such that the upper convex density of $E$ at this point is less than 1? In this paper, the authors construct a self-similar set satisfying the open set condition, which has a relative interior point with upper convex density less than 1. Thereby, the above problem is sufficiently answered.