Discrete & Computational Geometry

, Volume 7, Issue 3, pp 295–318

Quantitative Steinitz's theorems with applications to multifingered grasping

  • David Kirkpatrick
  • Bhubaneswar Mishra
  • Chee-Keng Yap
Article

DOI: 10.1007/BF02187843

Cite this article as:
Kirkpatrick, D., Mishra, B. & Yap, CK. Discrete Comput Geom (1992) 7: 295. doi:10.1007/BF02187843

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.

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • David Kirkpatrick
    • 1
  • Bhubaneswar Mishra
    • 2
  • Chee-Keng Yap
    • 2
  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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