EDB — 16Z

Exercises

1. [16Z] Topics:simplex.

   Given $x_0,\dots x_k\in {ℝ}^n$, let

   $$\begin{array}{}\text{(1)}& \{{\sum }_{i=0}^k{x}_i{t}_i:{\sum }_{i=0}^k{t}_i=1\forall i,t_i\ge 0\}\end{array}$$

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

   When the vectors $x_1-x_0,x_2-x_0\dots x_k-x_0$ 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

   $$\begin{array}{}\text{(2)}& \{{\sum }_{i=0}^n{x}_i{t}_i:{\sum }_{i=0}^n{t}_i=1\forall i,t_i>0\}\end{array}$$

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