Finite and infinitesimal rigidity with polyhedral norms

TL;DR

This paper characterizes finite and infinitesimal rigidity of bar-joint frameworks in Euclidean space with respect to polyhedral norms, establishing equivalences and combinatorial criteria, including an analogue of Laman's theorem for the plane.

Contribution

It provides a comprehensive characterization of rigidity under polyhedral norms, extending classical results and introducing new combinatorial conditions.

Findings

Infinitesimal and continuous rigidity are equivalent for well-positioned frameworks.

Rigidity can be characterized using monochrome spanning trees based on edge labelling.

An analogue of Laman's theorem is established for polyhedral norms in two dimensions.

Abstract

We characterise finite and infinitesimal rigidity for bar-joint frameworks in R^d with respect to polyhedral norms (i.e. norms with closed unit ball P a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in R^d which are well-positioned with respect to P. An edge-labelling determined by the facets of the unit ball and placement of the framework is used to characterise infinitesimal rigidity in R^d in terms of monochrome spanning trees. An analogue of Laman's theorem is obtained for all polyhedral norms on R^2.