On the Impossibility of Universal Ptolemaic Normalized Metrics

Abstract

We prove that no continuous radial weight function can universally induce a normalized metric satisfying Ptolemy's inequality across all normed spaces. This resolves a fundamental aspect of the Klamkin-Meier problem by demonstrating that radial rescaling, while effective within specific geometric classes, cannot overcome the structural heterogeneity of Banach spaces: non-inner-product spaces lack the intrinsic Ptolemaic structure that radial rescaling require. While we characterize when such metrics exist via M-Ptolemaic subadditivity, universality is achievable only within the class of inner product spaces where the Ptolemaic structure is already inherent to the geometry.