Best Possible Triangle Inequalities for Statistical Metric Spaces1

1. A statistical metric space2 (briefly, an S M space) is a set 5 and a mapping $ from SXS into the set of distribution functions (i.e., real-valued functions of a real variable which are everywhere defined, nondecreasing, left-continuous and have inf 0 and sup 1). The distribution function *5(p, q) associated with a pair of points (p, q) in S is denoted by Fpq. The functions Fpq are assumed to satisfy: (SM-I) Fpq(x) = 1 for all x > 0 iff p = q. (SM-II) Fpq(0)=0. (SM-III) FPq = Fqp. (SM-IV) If Fpt(x) = 1 and Fqr(y) = 1, then Fpr(x+y) = 1. A real-valued function T, whose domain is the set of real number pairs (x, y) such that 0 5¡x, y SI, is called a t-function if it satisfies the following conditions: (T-I) T(a, l)=a, T(0, 0)=0. (T-II) T(c, d) ^ T(a, b) if c^a, d^b. (T-III) T(a,b) = T(b,a) (commutativity). (T-IV) T[T(a, b), c] = T[a, T(b, c)] (associativity). Definition 1. A Menger space (S, 5, T) is an SM space (S, ff) and a ¿-function T such that the triangle inequality, (SM-IVm) Fpr(x+y)^T(Fpq(x), Fqr(y)), holds for all points p, q, r in S and for all numbers x, y^O. Definition 2. The ¿-function Ti is stronger than the ¿-function T2, and we write Ti^T2, if 7"i(x, y) ^ T2(x, y) for O^x, yál¡ Ti is strictly stronger than T2 if Ti is stronger than T2 and there is at least one pair of numbers (x, y) such that 7\(x, y) > T2(x, y). Correspondingly, T2 is weaker or strictly weaker than Tu For a given SM space there is in general more than one ¿-function which makes it a Menger space. In particular, if (S, í, J") is a Menger space and U is weaker than T, then (S, ï, U) is also a Menger space