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On the monotonicity of magnitude functions of negative type finite semimetric spaces
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.
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BibTeX
@misc{gomi2026monotonicity,
title = {{On the monotonicity of magnitude functions of negative type finite semimetric spaces}},
author = {Kiyonori Gomi and Hiroshi Tsuji},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.15720},
eprint = {2608.15720},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15720},
}
RIS
TY - UNPB
TI - On the monotonicity of magnitude functions of negative type finite semimetric spaces
AU - Kiyonori Gomi
AU - Hiroshi Tsuji
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2608.15720
UR - https://arxiv.org/abs/2608.15720
AB - 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.
ER -