Intersection of random planes

  • Thread starter Dragonfall
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  • #1
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Main Question or Discussion Point

Let [itex]x \in \{-1, 1\}^n[/itex] and let [itex]p(x) = \{w \in \mathbb{R}^n : x \cdot w > 1\}[/itex]. What is the probability that [itex]p(x_1) \cap \ldots \cap p(x_{n+1}) = \emptyset[/itex] given that [itex]x_i[/itex] are chosen uniformly at random?
 

Answers and Replies

  • #2
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##p(x)## is not a plane. It is a half-space. If you insist on the symbolical formulation, then the question boils down to the probability of having at least two vectors in ## \{ x_1, \ ... \ , x_{n + 1} \} ## that are anti-parallel.
 
  • #3
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No, it's possible to have empty intersection without a pair of opposite vectors.
 

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