Refined inequalities for the distance in metric spaces

Abstract

In this note we prove among others that ∑1≤i<j≤npipjdS(xi,xj)≤ {2s−1 infx∈X [ Σ k=1n pk (1 − pk) ds (xk, x)], s ≥ 1; infx∈X [ Σ k=1n pk (1 − pk) ds (xk, x)] , 0 < s < 1, where (X, d) is a metric space, xi ∈ X, pi ≥ 0, i ∈ {1, ..., n} with Σ i=1n pi = 1 and s > 0. This generalizes and improves some early upper bounds for the sum Σ1≤i<j≤n pipjd (xi, xj) .