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Math Help - Convex hull

  1. #1
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    Convex hull

    Hello,

    I am not sure I am posting in the right section so please tell me if I need to move this thread.

    I am facing a quite difficult problem. Here it is:

    Let \pi_0,...,\pi_m \in S_l=\{x\in R_+^l, \quad \sum_{h=1}^lx_h=1\}.
    Let us denote by \Sigma=co\{\pi_0,...,\pi_m\}=\{\sum_{i=0}^m\lambda  _i\pi_i, \quad (\lambda_i)_{i=0,...,m}\in S_{m+1}\} the convex hull of the points \pi_0,...,\pi_m.
    K=\{\pi_{i_1},...,\pi_{i_k}\} is the set of extremal points of \Sigma (ie those at the boundary of the convex hull) and \partial I = \{i_1,...,i_k\} are the indexes of those points.
    Finally, we define the correspondence \phi_p: i \to \phi_p(i) = \{j\in\{1,...,l\}, \quad \frac{\pi_{ij}}{p_j} = \max_{1\leq h \leq l} \frac{\pi_{ih}}{p_h}\} where p\in\Sigma.

    It is only a guess, but I think that \cup_{i\in\partial I} \phi_p(i) equals either \{1,...,l\}, or \{1,...,l\}-\{v\} where index v is such that p_v = \max_{1\leq h \leq l} p_h.

    If you have any hint on this topic, please let me know !

    Thanks.
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  2. #2
    Senior Member
    Joined
    Mar 2010
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    Beijing, China
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    Re: Convex hull

    sorry, just understood the problem.
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