The Assouad dimension of , is the infimum of all such that is -homogeneous for some .[3]
Let be a metric space, and let E be a non-empty subset of X. For r > 0, let denote the least number of metric open balls of radius less than or equal to r with which it is possible to cover the set E. The Assouad dimension of E is defined to be the infimal for which there exist positive constants C and so that, whenever
the following bound holds:
The intuition underlying this definition is that, for a set E with "ordinary" integer dimension n, the number of small balls of radius r needed to cover the intersection of a larger ball of radius R with E will scale like (R/r)n.
The Assouad dimension of a metric space is always greater than or equal to its upper box dimension, which in turn is greater than or equal to the Hausdorff dimension.[5]
The Lebesgue covering dimension of a metrizable spaceX is the minimal Assouad dimension of any metric on X. In particular, for every metrizable space there is a metric for which the Assouad dimension is equal to the Lebesgue covering dimension.[5]
References
^Assouad, Patrice (1979). "Étude d'une dimension métrique liée à la possibilité de plongements dans Rn". Comptes Rendus de l'Académie des Sciences, Série A-B (in French). 288 (15): A731–A734. ISSN0151-0509. MR532401