Connecting Hausdorff measure and upper convex density or Hs-a.e. covering ✩

In this paper, we make precise the relationship between Hausdorff measure and upper convex density or Hs -a.e. covering. We mainly study a class of s-sets, namely, the s-straight s-sets, especially the self-similar s-sets, which are the most important fractals in the study of fractal geometry. Sufficient and necessary conditions for the s-straight s-sets associated with Hs -a.e. covering, for the upper convex density of the self-similar s-set at the simple-contracting-similarity fixed point less than 1, and for the existence of the best Hs -a.e. covering of the self-similar s-set, are obtained. In addition, some results are applied to the famous classical fractals.  2005 Elsevier Inc. All rights reserved