On the $p$-negative type gap of finite metric spaces and its relation to the Gramian matrix

Abstract:

The $p$-negative type gap of a finite metric space is a useful tool in studying isometric embedding properties of the space. Wolf gave a versatile formula for computing the $p$-negative type gap, which relies on properties of the inverse of the matrix $D_{p}=(d_{X}(x_{i},x_{j})^{p})_{i,j=0}^{n}$. In this article we provide a simplification of Wolf’s formula in terms of the Gramian matrix $G_{p}=((1/2)(d_{X}(x_{i},x_{0})^{p}+d_{X}(x_{j},x_{0})^{p}-d_{X}(x_{i},x_{j})^{p}))_{i,j=1}^{n}$. We also study slight variations of the $p$-negative type gap and show that that some of these are much easier to compute than the original $p$-negative type gap. Finally, we provide explicit expressions for the $p$-negative type gap and some of these variations for the bipartite graph $K_{n,1}$.