Estimating the gap of finite metric spaces of strict \(p\)-negative type (Q1668994)

This paper considers finite metric spaces of strict \(p\)-negative type. It is shown that strict \(p\)-negativity is equivalent to the \(p\)-distance matrix \(D_p\) having one positive eigenvalue with all other eigenvalues negative and satisfying the inequality \(\langle D_p u , D_p \rangle > 0\) where \(u\) is the vector with every entry equal to 1. Setting \(M_p = \langle D_p u , D_p \rangle ^{-1}\), bounds for the so-called `gap' of the space are given. Upper bounds are given in terms of \(D_p\), in terms of the diameter of the space and interms of the second largest eigenvalue. A lower bound is given in terms of \(M_p\) and the top two eigenvalues. The gap is estimated under a certain gluing construction.