A variance inequality ensuring that a pre-distance matrix is Euclidean (Q2496617)

Consider real symmetric \(n\times n\) matrices \(D=(d_{ij})\), in which \(d_{ii}=0 \) and \(d_{ij}>0\) for \(i\neq j\). If there exist points \(p_1, \dots, p_n\in \mathbb R^n\) such that \(d_{ij}\) are precisely the squares of the Euclidean distances between the pair \(p_i\) and \(p_j\), \(D\) is called a Euclidean distance matrix. Note that for a given \(x>0\), the matrix \(D_x\) in which \(d_ {ij}=x\) for \(i\neq j\) is Euclidean, and by a continuity argument any \(D\) sufficiently close to \(D_x\) is Euclidean. Using these observations, conditions on the variances of \(d_{ij}\) are given in this paper that ensure \(D\) is Euclidean. A typical result states that if there exists \(x>0\) such that \[ x^2-\sum_{i\neq j}(x-d_{ij})^2>0, \] then \(D=(d_{ij})\) is Euclidean.