Inmathematics, thepacking 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 sensedual toHausdorff dimension, since packing dimension is constructed by "packing" smallopen balls inside the given subset, whereas Hausdorff dimension is constructed by covering the given subset by such small open balls. Thepacking dimension was introduced by C. Tricot Jr. in 1982.

Let (X, d) be a metric space with a subsetS ⊆ X and lets ≥ 0 be a real number. Thes-dimensional packing pre-measure ofS is defined to be

${\displaystyle P_0^s(S)=\underset{\delta \downarrow 0}{lim sup}\left\{\sum _{i\in I}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(B_i{)}^s|\begin{array}{c}\{B_i{\}}_{i\in I}\text{ is a countable collection}\\ \text{of pairwise disjoint closed balls with}\\ \text{diameters }\le \delta \text{ and centres in }S\end{array}\right\}.}$

Unfortunately, this is just apre-measure and not a truemeasure on subsets ofX, as can be seen by consideringdense,countable subsets. However, the pre-measure leads to abona fide measure: thes-dimensional packing measure ofS is defined to be

${\displaystyle P^s(S)=inf\left\{\sum _{j\in J}{P}_0^s(S_j)|S\subseteq \bigcup _{j\in J}{S}_j,J\text{ countable}\right\},}$

i.e., the packing measure ofS is theinfimum of the packing pre-measures of countable covers ofS.

Having done this, thepacking dimension dim_P(S) ofS is defined analogously to the Hausdorff dimension:

The following example is the simplest situation where Hausdorff and packing dimensions may differ.

Fix a sequence${\displaystyle (a_n)}$ such that${\displaystyle a_0=1}$ and. Define inductively a nested sequence${\displaystyle E_0\supset E_1\supset E_2\supset \cdots }$ of compact subsets of the real line as follows: Let${\displaystyle E_0=[0,1]}$. For each connected component of${\displaystyle E_n}$ (which will necessarily be an interval of length${\displaystyle a_n}$), delete the middle interval of length${\displaystyle a_n-2a_{n+1}}$, obtaining two intervals of length${\displaystyle a_{n+1}}$, which will be taken as connected components of${\displaystyle E_{n+1}}$. Next, define${\displaystyle K=\bigcap _n{E}_n}$. Then${\displaystyle K}$ is topologically a Cantor set (i.e., a compact totally disconnected perfect space). For example,${\displaystyle K}$ will be the usual middle-thirds Cantor set if${\displaystyle a_n=3^{-n}}$.

It is possible to show that the Hausdorff and the packing dimensions of the set${\displaystyle K}$ are given respectively by:

It follows easily that given numbers${\displaystyle 0\le d_1\le d_2\le 1}$, one can choose a sequence${\displaystyle (a_n)}$ as above such that the associated (topological) Cantor set${\displaystyle K}$ has Hausdorff dimension${\displaystyle d_1}$ and packing dimension${\displaystyle d_2}$.

One can considerdimension functions more general than "diameter to thes": for any functionh : [0, +∞) → [0, +∞], let thepacking pre-measure ofSwith dimension functionh be given by

${\displaystyle P_0^h(S)=\underset{\delta \downarrow 0}{lim}sup\left\{\sum _{i\in I}h(\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(B_i))|\begin{array}{c}\{B_i{\}}_{i\in I}\text{ is a countable collection}\\ \text{of pairwise disjoint balls with}\\ \text{diameters }\le \delta \text{ and centres in }S\end{array}\right\}}$

and define thepacking measure ofSwith dimension functionh by

${\displaystyle P^h(S)=inf\left\{\sum _{j\in J}{P}_0^h(S_j)|S\subseteq \bigcup _{j\in J}{S}_j,J\text{ countable}\right\}.}$

The functionh is said to be anexact (packing)dimension function forS ifP^h(S) is both finite and strictly positive.

• IfS is a subset ofn-dimensionalEuclidean spaceRⁿ with its usual metric, then the packing dimension ofS is equal to the upper modified box dimension ofS:$${\displaystyle {\dim }_{\mathrm{P}}(S)={\overline{\dim }}_{\mathrm{M}\mathrm{B}}(S).}$$ This result is interesting because it shows how a dimension derived from a measure (packing dimension) agrees with one derived without using a measure (the modified box dimension).

Note, however, that the packing dimension isnot equal to the box dimension. For example, the set ofrationalsQ has box dimension one and packing dimension zero.

• Hausdorff dimension
• Minkowski–Bouligand dimension

• Tricot, Claude Jr. (1982). "Two definitions of fractional dimension".Mathematical Proceedings of the Cambridge Philosophical Society.91 (1):57–74.doi:10.1017/S0305004100059119.S2CID 122740665.MR 0633256

• Dimension theory
• Fractals
• Metric geometry

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