Volume 26, Issue 4
A Note on Upper Convex Density

Commun. Math. Res., 26 (2010), pp. 361-368.

Published online: 2021-05

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• Abstract

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.

• Keywords

self-similar set, open set condition, upper convex density.

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@Article{CMR-26-361, author = {Jiandong and Yin and and 19178 and and Jiandong Yin and Zouling and Zhou and and 18278 and and Zouling Zhou}, title = {A Note on Upper Convex Density}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {26}, number = {4}, pages = {361--368}, abstract = {

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.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19132.html} }
TY - JOUR T1 - A Note on Upper Convex Density AU - Yin , Jiandong AU - Zhou , Zouling JO - Communications in Mathematical Research VL - 4 SP - 361 EP - 368 PY - 2021 DA - 2021/05 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19132.html KW - self-similar set, open set condition, upper convex density. AB -

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.

Jiandong Yin & Zouling Zhou. (2021). A Note on Upper Convex Density. Communications in Mathematical Research . 26 (4). 361-368. doi:
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