# Minimizing geodesic nets and critical points of distance

## TL;DR

This paper explores the relationship between geodesic nets and critical points of the distance function, providing bounds on the number of balanced points and length of minimizing geodesic nets on spheres.

## Contribution

It establishes new bounds on the number of balanced points and the length of minimizing geodesic nets on manifolds homeomorphic to spheres.

## Key findings

- Bounded the number of balanced points for certain geodesic nets.
- Bounded the length of certain minimizing geodesic nets.
- Linked geodesic nets to critical points of the distance function.

## Abstract

In this paper we establish a relationship between geodesic nets and critical points of the distance function. We bound the number of balanced points for certain minimizing geodesic nets on manifolds homeomorphic to the $n$-sphere. We also bound the length of certain minimizing geodesic nets.