Random polytopes in the d-dimensional cube

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23 Citations (Scopus)

Abstract

Let Cd be the set of vertices of a d-dimensional cube, Cd={(x1, ..., xd):xi=±1}. Let us choose a random n-element subset A(n) of Cd. Here we prove that Prob (the origin belongs to the conv A(2 d+x→2 d))=φ(x)+o(1) if x is fixed and d → ∞. That is, for an arbitrary ε>0 the convex hull of more than (2+ε)d vertices almost always contains 0 while the convex hull of less than (2-ε)d points almost always avoids it.

Original languageEnglish
Pages (from-to)315-319
Number of pages5
JournalDiscrete & Computational Geometry
Volume1
Issue number1
DOIs
Publication statusPublished - Dec 1 1986

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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