caa a22 4500 469075236 CHVBK 20180323132933.0 cr unu---uuuuu 170328e19920301xx s 000 0 eng 10.1007/BF02187843 doi (NATIONALLICENCE)springer-10.1007/BF02187843 Quantitative Steinitz's theorems with applications to multifingered grasping [Elektronische Daten] [David Kirkpatrick, Bhubaneswar Mishra, Chee-Keng Yap] 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 % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeei0xd9Wq-Jb9% vqaqFfpe0db9as0-LqLs-Jirpepeei0-bs0Fb9pgea0db9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaigdacqGHsi% slcaaIZaGaamizamaabmaabaWaaSaaaeaacaaIYaGaamizamaaCaaa% leqabaGaaGOmaaaaaOqaaiaad2gaaaaacaGLOaGaayzkaaWaaWbaaS% qabeaacaaIYaGaai4laiaacIcacaWGKbGaeyOeI0IaaGymaiaacMca% aaGccaGGUaaaaa!4694! $$1 - 3d\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ 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 % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrpepeei0xd9Wq-Jb9% vqaqFfpe0db9as0-LqLs-Jirpepeei0-bs0Fb9pgea0db9fr-xfr-x% frpeWZqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaigdacqGHsi% sldaWcaaqaaiaaigdaaeaacaaIXaGaaG4naaaadaqadaqaamaalaaa% baGaaGOmaiaadsgadaahaaWcbeqaaiaaikdaaaaakeaacaWGTbaaaa% GaayjkaiaawMcaamaaCaaaleqabaGaaGOmaiaac+cacaGGOaGaamiz% aiabgkHiTiaaigdacaGGPaaaaOGaaiOlaaaa!4735! $$1 - \frac{1}{{17}}\left( {\frac{{2d^2 }}{m}} \right)^{2/(d - 1)} .$$ 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. Springer-Verlag New York Inc., 1992 Kirkpatrick David Department of Computer Science, University of British Columbia, Vancouver, Canada aut Mishra Bhubaneswar Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA aut Yap Chee-Keng Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA aut Discrete & Computational Geometry Springer New York 7/3(1992-03-01), 295-318 0179-5376 7:3<295 1992 7 454 https://doi.org/10.1007/BF02187843 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02187843 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Kirkpatrick David Department of Computer Science, University of British Columbia, Vancouver, Canada aut NATIONALLICENCE 700 1- Mishra Bhubaneswar Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA aut NATIONALLICENCE 700 1- Yap Chee-Keng Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA aut NATIONALLICENCE 773 0- Discrete & Computational Geometry Springer New York 7/3(1992-03-01), 295-318 0179-5376 7:3<295 1992 7 454 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer