, Volume 1, Issue 1, pp 315-319

Random polytopes in thed-dimensional cube

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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.