Finite and Infinitesimal Rigidity with Polyhedral Norms

Abstract

We characterise finite and infinitesimal rigidity for bar-joint frameworks in (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) in terms of monochrome spanning trees. An analogue of Laman’s theorem is obtained for all polyhedral norms on (Formula presented.).