Refined inequalities for the distance in metric spaces

Abstract

In this note we prove among others that

∑1≤i<j≤np_ip_jd^S(x_i,x_j)
≤ {2^s−1 inf_x∈X [ Σ _k=1ⁿ p_k (1 − p_k) d^s (x_k, x)], s ≥ 1; infx∈X [ Σ _k=1ⁿ p_k (1 − p_k) d^s (x_k, x)] , 0 < s < 1, where (X, d) is a metric space, x_i ∈ X, p_i ≥ 0, i ∈ {1, ..., n} with Σ _i=1ⁿ p_i = 1 and s > 0. This generalizes and improves some early upper bounds for the sum Σ_1≤i<j≤n p_ip_jd (x_i, x_j) .