The strength of a geometric simplex
Metric Geometry
2026-04-07 v2
Abstract
The basic input for many real objects is a finite cloud of unordered points. The strongest equivalence between objects in practice is rigid motion in a Euclidean space. A recent polynomial-time classification of point clouds required a Lipschitz continuous function that vanishes on degenerate simplices, while the usual volume is not Lipschitz. We define the strength of any geometric simplex and prove its continuity under perturbations with explicit bounds for Lipschitz constants.
Cite
@article{arxiv.2602.17630,
title = {The strength of a geometric simplex},
author = {Olga Anosova and Vitaliy Kurlin},
journal= {arXiv preprint arXiv:2602.17630},
year = {2026}
}
Comments
11 pages, 2 figures. The second version extended Theorem 2.4(b) to the signed strength and streamlined the proofs. The latest version is maintained at https://kurlin.org/projects/cloud-isometry-spaces/strength-simplex.pdf