# Approximating the Convex Hull via Metric Space Magnitude

## TL;DR

This paper introduces a novel method to approximate convex hulls of finite Euclidean point sets using the concept of magnitude from metric space theory, providing explicit formulas and an intrinsic point ordering.

## Contribution

It develops an explicit corrected inclusion-exclusion formula and defines a new point-based quantity called the moment, enabling a convex hull approximation algorithm.

## Key findings

- Derived an explicit formula for the corrected inclusion-exclusion principle.
- Defined the moment to intrinsically order points within a metric space.
- Created an algorithm that approximates convex hulls using magnitude-based concepts.

## Abstract

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of effective species in an environment; when applied to a one-parameter family of metric spaces $tX$ with scale parameter $t$, the magnitude captures much of the underlying geometry of the space. Prior work has mostly focussed on properties of magnitude in a global sense; in this paper we restrict the sets to finite subsets of Euclidean space and investigate its individual components. We give an explicit formula for the corrected inclusion-exclusion principle, and define a quantity associated with each point, called the $\textit{moment}$ which gives an intrinsic ordering to the points. We exploit this in order to form an algorithm which approximates the convex hull.