# is there anything intelligent to say about the following span of vectors?

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!

-
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. – Arturo Magidin May 8 '11 at 22:59
$v_i$ is a vector, so $v_{ij}$ is the $j$th coordinate of vector $v_i$. – kopi May 8 '11 at 23:01
not every "vector" in a vector space is a tuple of entries. It would be best to say so explicitly. – Arturo Magidin May 8 '11 at 23:06
that is correct. I changed it to "real vectors". thanks. – kopi May 8 '11 at 23:08

The vectors $v_i$ are arbitrary points in the simplex of the unknown dimension of the space (since you write sums I will assume that we are in an $\mathbb R^m$).
Certainly, the convex sets live at most in affine dimension $n-1$ and they have at most $n$ vertices, but they can be smaller and have less vertices if some vectors are in the span of other vectors.