Neighborly Embedded Manifolds |
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| Authors: | G Kalai A Wigderson |
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| Institution: | (1) Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel;(2) Departments of Computer Science and Mathematics, Yale University, New Haven, USA;(3) Department of Mathematics, Institute for Advanced Studies, Princeton, USA |
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| Abstract: | An embedding of an n-dimensional manifold M into R d is called k-neighborly if, for every k points on the embedded manifold, there is a hyperplane H in R d which supports the manifold precisely at these points. Micha A. Perles (Problems presented in Oberwolfach conference on “Convexity”, 1982]) asked: What is the smallest dimension d(k,n) of the ambient space in which a k-neighborly n-dimensional manifold exists? We prove that d(k,n)≤2k(k?1)n. Related results and open problems are discussed. |
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| Keywords: | Convex bodies Polytopes Neighborliness Cyclic polytopes Continuous hashing |
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