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The Convex Hull of Random Hyperplanes
Let $\pi_1,\ldots,\pi_n$ be hyperplanes in general position in $R^d$, $V=\{\pi_{j_1}\cap\cdots \cap \pi_{j_d}, 1\leq j_1<j_2 < \cdots
<j_d\leq n\}$ the set of vertices, and $N=\vert$conv$(V)\vert$, the number of extreme points of the convex hull of $V$. We show that for $d\leq 3$, if the $\pi_i$ are chosen ``randomly'', there is a constant $c_d>0$ depending on $d$ but not on $n$ so that $E(N)<c_d$. The prospects for $d>3$ will be discussed.

Adrian Vetta 2004-01-20