Abstract
LetC d be the set of vertices of ad-dimensional cube,C d={(x 1, ...,x d ):x i =±1}. Let us choose a randomn-element subsetA(n) ofC d. Here we prove that Prob (the origin belongs to the convA(2d+x→2d))=φ(x)+o(1) ifx is fixed andd → ∞. That is, for an arbitraryε>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.
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Füredi, Z. Random polytopes in thed-dimensional cube. Discrete Comput Geom 1, 315–319 (1986). https://doi.org/10.1007/BF02187704
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DOI: https://doi.org/10.1007/BF02187704