The strength of a geometric simplex

TL;DR

This paper introduces a new measure called the strength of a geometric simplex, proving its continuity under perturbations and providing explicit bounds, which enhances the classification of point clouds in Euclidean space.

Contribution

It defines the strength of geometric simplices and establishes its continuity with explicit bounds, improving robustness in point cloud classification.

Findings

The strength measure is Lipschitz continuous under perturbations.

Explicit bounds for Lipschitz constants are provided.

The measure aids in classifying point clouds with geometric invariance.

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.