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Let's say I have a set of real vectors $v_1,\ldots,v_n$ such that $\sum_j v_{ij} = 1$ for all $i$ and $v_{ij} \ge 0$.

Now consider the set $\Gamma(n) = \{ \beta \mid \sum_i \beta_i = 1, \beta_i \ge 0 \}$, i.e. the set of vectors of dimension $n$ in the probability simplex.

Is there anything interesting to say about the span $\{ \sum_i \beta_i v_i \mid \beta \in \Gamma(n) \}$?

Under all kind of different conditions... let's say $v_i$ are independent, or that $n$ is larger than the length of each $v_i$, or anything at all. I am trying to see what properties I can have from such a span.

Thanks!

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    What does "$v_{ij}$" mean? Is each vector a tuple, $v_i=(v_{i1},v_{i2},\ldots,v_{in})$? If so, you might want to say so.2011-05-08
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    $v_i$ is a vector, so $v_{ij}$ is the $j$th coordinate of vector $v_i$.2011-05-08
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    not every "vector" in a vector space is a tuple of entries. It would be best to say so explicitly.2011-05-08
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    that is correct. I changed it to "real vectors". thanks.2011-05-08

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