# What happens if i cluster data with a distance metric, that is not a distance metric?

i stumbled upon a paper, that introduces a distance metric, which is then used to cluster data (https://doi.org/10.1137/1.9781611972795.35). I noticed however, that this "distance" violated at least one property of the mathematical definition of distance, namely $$dist(x,x) \neq 0$$ for some $$x$$. Additionally, I am pretty sure, that $$dist(x,y) < 0$$ for some $$x,y$$

I was wondering, what could go wrong when clustering data if theses properties of distances are violated? And what happens if other properties are violated such as symmetry or triangle inequality?