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Equivalent Metrics


Two metrics g_1 and g_2 defined on a space X are called equivalent if they induce the same metric topology on X. This is the case iff, for every point x_0 of X, every ball with center at x_0 defined with respect to g_1:

 B(x_0,r_1;g_1)={x in X|g_1(x_0,x)<r_1}
(1)

contains a ball with center x_0 with respect to g_2:

 B(x_0,r_2;g_2)={x in X|g_2(x_0,x)<r_2},
(2)

and conversely.

Every metric g on X has uncountably many equivalent metrics. For every positive real number epsilon, a "scaled" metric g_epsilon can be defined such that for all x,y in X,

 g_epsilon(x,y)=(g(x,y))/epsilon.
(3)

In fact, for all x_0 in X:

 B(x_0,r;g)=B(x_0,r/epsilon;g_epsilon).
(4)

Another metric g^' equivalent to g is defined by

 g^'(x,y)=(g(x,y))/(1+g(x,y)),
(5)

for all x,y in X. In fact,

 B(x_0,r;g) subset= B(x_0,r;g^'),
(6)

and

 B(x_0,r/(r+1);g^') subset= B(x_0,r;g).
(7)

In the Euclidean plane R^2, the metric

 g((x_1,y_1),(x_2,y_2))=sqrt((x_1-x_2)^2+(y_1-y_2)^2),
(8)

with circular balls can be defined in addition to the Euclidean metric. An equivalent more general metric for all positive real numbers a and b can be defined as

 g_(a,b)((x_1,y_1),(x_2,y_2))=sqrt(((x_1-x_2)^2)/(a^2)+((y_1-y_2)^2)/(b^2)),
(9)

with elliptic balls, and the taxicab metric

 g_t((x_1,y_1),(x_2,y_2))=|x_1-x_2|+|y_1-y_2|,
(10)

can be defined with square "balls." All these are equivalent to the Euclidean metric.


See also

Metric, Metric Equivalence Problem, Product Metric

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Equivalent Metrics." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/EquivalentMetrics.html

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