The CW-Inequalities for Vectors in l₁

Let l₁ denote the metric space the elements of which are all sequences x=(x₁, x₂,...) of real numbers satisfying ∑ i=1∞ |xi|<∞. The distance d(x, y) between x and y = (y₁, y₂,...) is d(x,y)=∑ i=1∞ |xi−yi|. Let υ₀, υ₁,..., υ_m−1, u₀, u₁,...,u_k-1 be a sequence of n = m + k not necessarily distinct vectors in l₁, where k > 0, m = k + 2u + 1 and u ⩾ 0. We show that ∑ 0⩽i<j<m (i−j)(modm)∈{±1,±2...,±u} d(υi,υj)+∑ 0⩽i<m 0⩽j<k d(υi,uj)⩾∑ 0⩽i<j<m d(υi,υj)+∑ 0⩽i<j<k d(ui,uj). This solves a conjecture of Deza and Laurent and implies, as special cases, inequalities of Deza, Laurent, Kelly, Barahona and Mahjoub.