Magnitude and matrix inequalities

Abstract: We present several applications of matrix-theoretic inequalities to the magnitude of metric spaces. We first resolve an open problem by showing that the magnitude of any finite metric space of negative type is less than or equal to its cardinality. This is a direct consequence of Styan's matrix inequality involving the Hadamard product of matrices. By related methods we also show a subadditivity property for the magnitude function of negative type compact metric spaces, and prove a convexity property for the magnitude for metrics interpolating in a natural way between two given, comparable metrics on a given set.