Metric dimension

A numerical characteristic of a compact set, defined in terms of coverings of "standard measure" , the number of which defines the metric dimension. Let $F$ be a compact set, and let $N _ {F} ( \epsilon )$ be the minimal number of sets with diameter not exceeding $\epsilon$ that are needed in order to cover $F$. This function, depending on the metric in $F$, takes integer values for all $\epsilon > 0$, and increases without bound as $\epsilon \rightarrow 0$; it is called the volume function of $F$. The metric order of the compact set $F$ is the number

$$k = fnnme \underline{lim} \ \left ( - \frac{ \mathop{\rm ln} N _ {F} ( \epsilon ) }{ \mathop{\rm ln} \epsilon } \right ) .$$

This quantity is not yet a topological invariant. Thus, the metric order of a curve in the sense of Jordan (cf. Line (curve)) with the Euclidean metric is equal to 1, but for a curve in the sense of Jordan passing through a perfect totally-disconnected set in $\mathbf R ^ {n}$ of positive measure, this value is equal to $n$. However, the greatest lower bound of the metric orders for all metrics on $F$( called the metric dimension) is equal to the Lebesgue dimension (the Pontryagin–Shnirel'man theorem, 1931, see [1]).

References

 [1] W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)

Metric dimension makes sense for non-compact separable metrizable spaces (using totally bounded metrics), and the Pontryagin–Shnirel'man theorem extends to them. This was shown by E. Szpilrajn-Marczewski. See [a2].

There are also other types of metric-dependent dimension functions.

One example is the Hausdorff dimension.

Another example is obtained by modifying the definition of the covering dimension $\mathop{\rm dim}$( see Dimension): If $( X , d )$ is a metric space, one defines $\mu \mathop{\rm dim} ( X , d )$ by $\mu \mathop{\rm dim} ( X , d ) \leq n$ if and only if for every $\epsilon > 0$ there is an open covering $\mathfrak U$ of $X$ with $\textrm{ mesh } \mathfrak U \leq n + 1$ and $\mathop{\rm ord} \mathfrak U < \epsilon$. Here $\textrm{ mesh } \mathfrak U = \sup \{ { \mathop{\rm diam} ( U) } : {U \in \mathfrak U } \}$ and $\mathop{\rm ord} \mathfrak U \leq n + 1$ means that no point of $X$ is an element of more than $n + 1$ elements of $\mathfrak U$. One can show that $\mu \mathop{\rm dim} ( X , d ) \leq \mathop{\rm dim} X \leq 2 \mu \mathop{\rm dim} ( X , d )$ and that these inequalities are best possible, see [a1].

References

 [a1] R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50 [a2] J.-I. Nagata, "Modern dimension theory" , Interscience (1965)
How to Cite This Entry:
Metric dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Metric_dimension&oldid=47828
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article