Equivalence of Metrics - Strong Equivalence

Strong Equivalence

Two metrics and are strongly equivalent if and only if there exist positive constants and such that, for every ,

In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in, rather than potentially different constants associated with each point of .

Strong equivalence of two metrics implies topological equivalence, but not vice versa. An intuitive reason why topological equivalence does not imply strong equivalence is that bounded sets under one metric are also bounded under a strongly equivalent metric, but not necessarily under a topologically equivalent metric.

All metrics induced by the p-norm, including the euclidean metric, the taxicab metric, and the Chebyshev distance, are strongly equivalent.

Even if two metrics are strongly equivalent, not all properties of the respective metric spaces are preserved. For instance, a function from the space to itself might be a contraction mapping under one metric, but not necessarily under a strongly equivalent one.