A Note on Upper Convex Density (Communications in Mathematical Research, 2021)
For a self-similar set $E$ satisfying the open set condition, upper convexdensity is an important concept for the computation of its Hausdorff measure, andit is well known that the set of relative interior points with upper convex density 1has a full Hausdorff measure. But whether the upper convex densities of $E$ at allthe relative interior points are equal to 1? In other words, whether there exists arelative interior point of $E$ such that the upper convex density of $E$ at this point isless than 1? In this paper, the authors construct a self-similar set satisfying the openset condition, which has a relative interior point with upper convex density less than1. Thereby, the above problem is sufficiently answered.