### Abstract

We prove the following quantitative form of a classical theorem of Steintiz: Let m be sufficiently large. If the convex hull of a subset S of Euclidean d-space contains a unit ball centered on the origin, then there is a subset of S with at most m points whose convex hull contains a solid ball also centered on the origin and having residual radius {Mathematical expression} The case m=2 d was first considered by Bárány et al. [1]. We also show an upper bound on the achievable radius: the residual radius must be less than {Mathematical expression} These results have applications in the problem of computing the so-called closure grasps by an m-fingered robot hand. The above quantitative form of Steinitz's theorem gives a notion of efficiency 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.

Original language | English (US) |
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Pages (from-to) | 295-318 |

Number of pages | 24 |

Journal | Discrete & Computational Geometry |

Volume | 7 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1992 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Discrete & Computational Geometry*,

*7*(1), 295-318. https://doi.org/10.1007/BF02187843