Your construction is a special case of semimetric spaces with relaxed triangle inequality: http://en.wikipedia.org/wiki/Semimetric_space#Semimetrics. This type of metric is sometimes also called non-Archimedian metric. There is a classic paper of W.A.Wilson "On semi-metric spaces", Amer. J. Math. 53 (1931) 361–373, on the subject. Also, I have seen this type of construction mostly used in fixed-point theory, so this would be an additional keyword to look for.
M.G.
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