---
abstract: 'A kissing sphere is a sphere that is tangent to a fixed reference ball. We develop in this paper a distance geometry for kissing spheres, which turns out to be a generalization of the classical Euclidean distance geometry.'
address: 'Freie Universität Berlin, Institut für Mathematik'
author:
- Hao Chen
bibliography:
- 'References.bib'
title: Distance Geometry for Kissing Spheres
---

[^1]

Introduction {#sec:Intro}
============

Distance geometry studies the geometry based only on knowledge of distances.

We develop in this paper a distance geometry for kissing spheres following the approach of Euclidean distance geometry. We first establish a distance space by defining a distance function (Section \[sec:Dist\]) on the set of kissing spheres. Then we study two basic problems of distance geometry: the embeddability problem (Section \[sec:Embed\]) and the distance completion problem (Section \[sec:Compl\]).

The key observation of this paper is that the distance matrix for kissing spheres also plays the role of Cayley-Menger matrix. It is then possible to adapt the proof techniques from Euclidean distance geometry for our use. Our main results (Theorem \[EmbK\] and \[CompK\]) are similar to the results in Euclidean distance geometry (Theorem \[EmbE\] and \[CompE\]). We also notice that the distance geometry for kissing spheres may degenerate to Euclidean distance geometry in different ways. In this sense, the distance geometry developped in this paper generalizes Euclidean distance geometry.

At the end of the paper, we will introduce some previous works on spheres, and point out the similarity and relations to our results.

Euclidean distance geometry {#sec:Euc}
---------------------------

Kissing spheres {#sec:KSphere}
---------------

Distance function {#sec:Dist}
=================

Embeddability problem {#sec:Embed}
=====================

Embedding into the lightcone {#sec:LCone}
============================

\[Lightcone\]

Distance completion problem {#sec:Compl}
===========================

Distance geometry for spheres {#sec:Sphere}
=============================

Acknowledgement {#acknowledgement .unnumbered}
===============

I would like to thank Günter M. Ziegler, my supervisor of PhD, for suggestions and careful review, thank Raman Sanyal for references, and thank Louis Theran and Karim Adiprasito for very helpful discussions.

[^1]: This research was supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408).
