Characterization of local dimension functions of subsets of ${ℝ}^d$

L. Olsen. "Characterization of local dimension functions of subsets of $ℝ^{d}$." Colloquium Mathematicae 103.2 (2005): 231-239. <http://eudml.org/doc/284206>.

@article{L2005,
abstract = {For a subset $E ⊆ ℝ^\{d\}$ and $x ∈ ℝ^\{d\}$, the local Hausdorff dimension function of E at x is defined by $dim_\{H,loc\}(x,E) = lim_\{r↘ 0\} dim_\{H\}(E ∩ B(x,r))$ where $dim_\{H\}$ denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.},
author = {L. Olsen},
journal = {Colloquium Mathematicae},
keywords = {Hausdorff dimension; packing dimension; dimension index; local Hausdorff dimension; local packing dimension},
language = {eng},
number = {2},
pages = {231-239},
title = {Characterization of local dimension functions of subsets of $ℝ^\{d\}$},
url = {http://eudml.org/doc/284206},
volume = {103},
year = {2005},
}

TY - JOUR
AU - L. Olsen
TI - Characterization of local dimension functions of subsets of $ℝ^{d}$
JO - Colloquium Mathematicae
PY - 2005
VL - 103
IS - 2
SP - 231
EP - 239
AB - For a subset $E ⊆ ℝ^{d}$ and $x ∈ ℝ^{d}$, the local Hausdorff dimension function of E at x is defined by $dim_{H,loc}(x,E) = lim_{r↘ 0} dim_{H}(E ∩ B(x,r))$ where $dim_{H}$ denotes the Hausdorff dimension. We give a complete characterization of the set of functions that are local Hausdorff dimension functions. In fact, we prove a significantly more general result, namely, we give a complete characterization of those functions that are local dimension functions of an arbitrary regular dimension index.
LA - eng
KW - Hausdorff dimension; packing dimension; dimension index; local Hausdorff dimension; local packing dimension
UR - http://eudml.org/doc/284206
ER -