- Packing dimension
In
mathematics , the packing dimension is one of a number of concepts that can be used to define thedimension of asubset of ametric space . Packing dimension is in some sense dual toHausdorff dimension , since packing dimension is constructed by "packing" smallopen ball s inside the given subset, whereas Hausdorff dimension is constructed by covering the given subset by such small open balls.Definitions
Let ("X", "d") be a metric space with a subset "S" ⊆ "X" and let "s" ≥ 0. The "s"-dimensional packing pre-measure of "S" is defined to be
:P_{0}^{s} (S) = lim_{delta downarrow 0} sup left{ left. sum_{i in I} mathrm{diam} (B_{i})^{s} ight| egin{matrix} { B_{i} }_{i in I} mbox{ is a countable collection} \ mbox{of pairwise disjoint balls with} \ mbox{diameters } leq delta mbox{ and centres in } S end{matrix} ight}.
Unfortunately, this is just a
pre-measure and not a true measure on subsets of "X", as can be seen by considering dense, countable subsets. However, the pre-measure leads to a "bona fide" measure: the "s"-dimensional packing measure of "S" is defined to be:P^{s} (S) = inf left{ left. sum_{j in J} P_{0}^{s} (S_{j}) ight| S subseteq igcup_{j in J} S_{j}, J mbox{ countable} ight},
i.e., the packing measure of "S" is the
infimum of the packing pre-measures of countable covers of "S".Having done this, the packing dimension dimP("S") of "S" is defined analogously to the Hausdorff dimension:
:egin{align}dim_{mathrm{P (S) &{} = sup { s geq 0 | P^{s} (S) = + infty } \&{} = inf { s geq 0 | P^{s} (S) = 0 }.end{align}
Generalizations
One can consider
dimension function s more general than "diameter to the "s": for any function "h" : [0, +∞) → [0, +∞] , let the packing pre-measure of "S" with dimension function "h" be given by:P_{0}^{h} (S) = lim_{delta downarrow 0} sup left{ left. sum_{i in I} h ig( mathrm{diam} (B_{i}) ig) ight| egin{matrix} { B_{i} }_{i in I} mbox{ is a countable collection} \ mbox{of pairwise disjoint balls with} \ mbox{diameters } leq delta mbox{ and centres in } S end{matrix} ight}
and define the packing measure of "S" with dimension function "h" by
:P^{h} (S) = inf left{ left. sum_{j in J} P_{0}^{h} (S_{j}) ight| S subseteq igcup_{j in J} S_{j}, J mbox{ countable} ight}.
The function "h" is said to be an exact (packing) dimension function for "S" if "P""h"("S") is both finite and strictly positive.
Properties
* If "S" is a subset of "n"-dimensional
Euclidean space R"n" with its usual metric, then the packing dimension of "S" is equal to the upper modified box dimension of "S":::dim_{mathrm{P (S) = overline{dim}_{mathrm{MB (S).
:This result is interesting because it shows how a dimension derived from a measure (packing dimension) agrees with one derived without using a measure (box dimension).
ee also
*
Hausdorff dimension
*Minkowski-Bouligand dimension References
* cite journal
last = Tricot, Jr.
first = Claude
title = Two definitions of fractional dimension
journal = Math. Proc. Cambridge Philos. Soc,
volume = 91
year = 1982
issue = 1
pages = 57–74
issn = 0305-0041 MathSciNet|id=633256
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