Key points are not available for this paper at this time.
The (length) excess of a triangle measures how much the triangle inequality fails to be an equality.This notion was first studied seriously in AG.We say that a (bounded) metric space X = (X, d) has excess :5 e if there are points p, q E X such that d(p, x)+d(x, q) ~ d(p, q)+e for all x EX.The excess, e(X) , of X is the infimum of all e ~ 0 where X has excess ~ e.
Grove et al. (Mon,) studied this question.