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

The p 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 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 Dₚ=(dX(xᵢ,xⱼ)ᵖ)i,j=0ⁿ . 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 Gₚ=((1/2)(dX(xᵢ,x₀)ᵖ+dX(xⱼ,x₀)ᵖ-dX(xᵢ,xⱼ)ᵖ))i,j=1ⁿ . We also study slight variations of the p p -negative type gap and show that that some of these are much easier to compute than the original p p -negative type gap. Finally, we provide explicit expressions for the p p -negative type gap and some of these variations for the bipartite graph K n , 1 Kn,1 .