, Volume 7, Issue 1, pp 295-318

Quantitative Steinitz's theorems with applications to multifingered grasping

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Abstract

We prove the following quantitative form of a classical theorem of Steintiz: Letm be sufficiently large. If the convex hull of a subsetS of Euclideand-space contains a unit ball centered on the origin, then there is a subset ofS with at mostm points whose convex hull contains a solid ball also centered on the origin and havingresidual radius $$1 - 3d\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ The casem=2d was first considered by Bárányet al. [1]. We also show an upper bound on the achievable radius: the residual radius must be less than $$1 - \frac{1}{{17}}\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ These results have applications in the problem of computing the so-calledclosure grasps by anm-fingered robot hand. The above quantitative form of Steinitz's theorem gives a notion ofefficiency for closure grasps. The theorem also gives rise to some new problems in computational geometry. We present some efficient algorithms for these problems, especially in the two-dimensional case.

This research was supported by NSF Grants DCR-84-01898, DCR-84-01633, ONR Grant N00014-89-J3042, and NSF Grant Subcontract CMU-406349-55586. Chee-Keng Yap was supported in part by the German Research Foundation (DFG) and by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM).