Immobilization of n-dimensional geometrical figures

Abstract

We treat for n-dimensional bodies the immobilization problem introduced by Kuperberg and Papadimitriou, giving the zerocth, first and second order condition for fixing, as well as a geometric characterization of the first order condition. We show the equivalence of the geometrical and mechanical conditions of fixing. We show that generically, C1 bodies may be fixed by n + 1 points. Also, there is a C1,1 neighbourhood of S in which a body not admitting threads (scts of trapping points which may slide along the surface) may be fixed by n + I points. We show that u star-shaped body trapped by a set P either may be fixed by a set similar to P, or admits a thread generated by P.