Infinitesimal rigidity and prestress stability for frameworks in normed spaces

TL;DR

This paper explores the rigidity of frameworks in normed spaces, introducing new concepts like prestress stability and second-order rigidity, and strengthening existing conditions using non-smooth analysis techniques.

Contribution

It extends the theory of framework rigidity to general normed spaces by defining prestress stability and second-order rigidity, and improves rigidity conditions through non-smooth analysis.

Findings

Strengthened sufficient conditions for framework rigidity in normed spaces.

Introduction of prestress stability and second-order rigidity concepts.

Application of non-smooth analysis to rigidity problems.

Abstract

A (bar-and-joint) framework is a set of points in a normed space with a set of fixed distance constraints between them. Determining whether a framework is locally rigid - i.e. whether every other suitably close framework with the same distance constraints is an isometric copy - is NP-hard when the normed space has dimension 2 or greater. We can reduce the complexity by instead considering derivatives of the constraints, which linearises the problem. By applying methods from non-smooth analysis, we shall strengthen previous sufficient conditions for framework rigidity that utilise first-order derivatives. We shall also introduce the notions of prestress stability and second-order rigidity to the topic of normed space rigidity, two weaker sufficient conditions for framework rigidity previously only considered for Euclidean spaces.