Year: 2026
Type: preprint
Abstract: Motivated by the conjectured monotonicity of the magnitude functions of metric spaces of negative type, we investigate the monotonicity problem for finite semimetric spaces, namely, ``metric spaces without the triangle inequality''. We completely resolve this problem by proving that the monotonicity holds for negative type semimetric spaces with at most three points, while constructing negative type semimetric spaces with at least four points whose magnitude functions are not monotonically increasing. We also prove monotonicity for all 4-point metric spaces and construct 12-point metric spaces of negative type whose magnitude functions are not monotonically increasing.
Source: arXiv (Cornell University)
Authors: Gomi, Kiyonori, Tsuji, Hiroshi
Institution: Institute of Science Tokyo
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203678004