Random polytopes in the d-dimensional cube

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. © 1986 Springer-Verlag New York Inc.