Quantitative Steinitz's Theorems with Applications to Multiingered Grasping 1 Section 1

We prove the following quantitative form of a classical theorem of Steinitz: Let m be suuciently 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 1 ? 3d 2d 2 m ! 2 d?1 : The case m = 2d was rst considered by BB arr any, Katchalski and Pach (1982). We also show an upper bound on the achievable radius: the residual radius must be less than 1 ? 1 17 2d 2 m ! 2 d?1 : These results have applications in the problem of computing the so-called closure grasps by an m-ngered robot hand. The above quantitative form of Steinitz's theorem gives a notion of eeciency for closure grasps. The theorem also gives rise to some new problems in computational geometry. We present some eecient algorithms for these problems, especially in the two dimensional case.