Quantitative Steinitz's theorems with applications to multifingered grasping (Q1182994)

The Steinitz theorem says that any point in the interior of the convex hull of a given subset \(S\) of a \(d\)-dimensional Euclidean space is also in the interior of the convex hull of some subset of \(S\) with at most \(2d\) points. \textit{I. Bárány, M. Katchalski}, and \textit{J. Pach} [Proc. Am. Math. Soc. 86, 109-114 (1982; Zbl 0511.52005)] proved a quantitative version of this theorem. The paper under review is concerned with a generalization of the Bárány result and with application to the problem of computing the closure grasp by \(m\)-fingered robot hand [the second author, \textit{J. T. Schwartz}, and \textit{M. Sharir}, Algorithmica 2, 541-558 (1987; Zbl 0652.70003)]. Also several new problems in computational geometry posed by this new theorem are studied.