Let d(x,y) be a metric on some space X, and let w(x) be a non-negative weight function defined on X. Define a new metric d2(x,y) on X such that d2(x,y) is monotonically increasing in d(x,y) and monotonically decreasing in w(x) and w(y).

This is a relatively open ended question. You may make further assumptions if it improves the simplicity, elegance, or usefulness (e.g., computational efficiency) of your solution.

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