Neighborly embedded manifolds

G. Kalai; A. Wigderson

doi:10.1007/s00454-008-9065-y

Neighborly embedded manifolds

G. Kalai^*
, A. Wigderson

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

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.

Original language	English
Pages (from-to)	319-324
Number of pages	6
Journal	Discrete and Computational Geometry
Volume	40
Issue number	3
DOIs	https://doi.org/10.1007/s00454-008-9065-y
State	Published - Oct 2008

Keywords

Continuous hashing
Convex bodies
Cyclic polytopes
Neighborliness
Polytopes

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AB - 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.

KW - Continuous hashing

KW - Convex bodies

KW - Cyclic polytopes

KW - Neighborliness

KW - Polytopes

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JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

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ER -

Neighborly embedded manifolds

Abstract

Keywords

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