# Finite and Infinitesimal Rigidity with Polyhedral Norms

Discrete & Computational Geometry, May 2015

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

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Derek Kitson. Finite and Infinitesimal Rigidity with Polyhedral Norms, Discrete & Computational Geometry, 2015, 390-411, DOI: 10.1007/s00454-015-9706-x