Packing dimension

In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimension, since packing dimension is constructed by 'packing' small open balls inside the given subset, whereas Hausdorff dimension is constructed by covering the given subset by such small open balls. The packing dimension was introduced by C. Tricot Jr. in 1982. In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimension, since packing dimension is constructed by 'packing' small open balls inside the given subset, whereas Hausdorff dimension is constructed by covering the given subset by such small open balls. The packing dimension was introduced by C. Tricot Jr. in 1982. 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 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 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: