Adding a point to configurations in closed balls

TL;DR

This paper characterizes when a new point can be added continuously to configurations of points in a closed ball, generalizing the Brouwer fixed-point theorem and revealing conditions based on the number and ordering of points.

Contribution

It provides a complete characterization of the conditions under which points can be added continuously to configurations in closed balls, extending classical fixed-point results.

Findings

A new point can be added continuously if and only if n ≠ 1 for ordered configurations.

For unordered configurations in dimension at least 2, a point can be added if and only if n = 2.

When n=2, the solution is unique up to homotopy.

Abstract

We answer the question of when a new point can be added in a continuous way to configurations of $n$ distinct points in a closed ball of arbitrary dimension. We show that this is possible given an ordered configuration of $n$ points if and only if $n \neq 1$. On the other hand, when the points are not ordered and the dimension of the ball is at least 2, a point can be added continuously if and only if $n = 2$. These results generalize the Brouwer fixed-point theorem, which gives the negative answer when $n=1$. We also show that when $n=2$, there is a unique solution to both the ordered and unordered versions of the problem up to homotopy.