If the triangle inequality does not hold for a distance function (i.e. it is not metric), will this limit its application in some area?
2026-04-01 20:04:44.1775073884
What are the limitations of non-metric distances?
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(This is a pretty quick-and-dirty answer. Probably a lot more work could go into making this either rigorous or intuitive, but I don't know the details.)
Yes, almost surely.
A function is positive-definite symmetric (PDS) if it is like a metric without triangle inequality. PDS functions are extremely common and so do not have a lot of unifying qualities. In particular, PDS functions do not necessarily induce a topology, and so almost all geometric meaning that can be associated with a metric cannot be transferred to a generic PDS function.