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Polytopes-The Dual Mapping

  1. Jan 13, 2012 #1
    1. The problem statement, all variables and given/known data

    Let v be a vertex of a d-polytope P such that [itex] 0 \in intP [/itex] .

    Prove that [itex] P^* \cap \{y \in \mathbb{R}^d \mid\left < y, v\right>=1\ \} [/itex] is a facet of [itex] P^{*} [/itex].


    2. Relevant equations

    The definitions are:

    [itex] P^*=\{ y\in\mathbb{R}^{d}\mid\left < x, y\right>\leq 1\ \forall x\in P\} [/itex]

    and a face of P is the empty set, P itself, or an intersection of P with a supporting hyperplane (i.e.- a hyperplane, such that P is located in one of the halfspaces it determines).

    A facet is a face of maximal degree

    3. The attempt at a solution
    I tried showing that if it isn't a facet (the fact that it's a face is obvious), we can delete one of the vertices that form [itex] P^{*} [/itex] but without any success.
  2. jcsd
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