# Critical configurations of solid bodies and the Morse theory of MIN functions

## TL;DR

This paper investigates the geometric configurations of solid bodies touching a central sphere, introducing the concept of critical clusters and analyzing their properties, with new findings on clusters of cylinders.

## Contribution

It introduces the notion of critical clusters for solid bodies and studies their properties, including new results on clusters of cylinders.

## Key findings

- Identification of critical clusters of balls and cylinders
- Establishment of criticality properties for these clusters
- Discovery of new critical clusters of cylinders

## Abstract

We study the manifold of clusters of nonintersecting congruent solid bodies, all touching the central ball $B\subset\mathbb{R}^{3}$ of radius one. Two main examples are clusters of balls and clusters of infinite cylinders. We introduce the notion of \textit{critical cluster} and we study several critical clusters of balls and of cylinders. For the case of cylinders some of our critical clusters are new. We also establish the criticality properties of clusters, introduced earlier by W. Kuperberg.