EDB — 16Z

Exercises

1. [16Z] Topics:simplex.

   Given \(x_ 0,\ldots x_ k∈ ℝ^ n\), let

   \begin{equation} \left\{ ∑_{i=0}^ k x_ i t_ i : ∑_{i=0}^ k t_ i=1 ∀ i, t_ i≥ 0 \right\} \label{eq:simplesso} \end{equation}

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   the set of all possible combinations: prove that this set is convex.

   When the vectors \(x_ 1-x_ 0,x_ 2-x_ 0\ldots x_ k-x_ 0\) are linearly independent, the set defined above is a simplex of dimension \(k\).

   Show that, if \(n=k\), then the simplex has a non-empty interior, equal to

   \begin{equation} \left\{ ∑_{i=0}^ n x_ i t_ i : ∑_{i=0}^ n t_ i=1 ∀ i, t_ i> 0 \right\} \label{eq:interno_ simplesso} \end{equation}

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