On the ๐-negative type gap of finite metric spaces and its relation to the Gramian matrix
Gavin Robertson ยท Proceedings of the American Mathematical Society ยท 2024
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 D_{p}=(d_{X}(x_{i},x_{j})^{p})_{i,j=0}^{n} . In this article we provide a simplification of Wolfโs 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 โ d X ( x i , x j ) p