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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.


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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. –  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

1 Answer 1

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 $\mathbb R^m$).

The span consists of all convex combination of these points, so you get all possible convex sets contained in the standard simplex.

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.

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you just made me realize that I defined a convex hull of points in the probability simplex. Is there any characterization for the convex hull of such a set based on some properties of the set? –  kopi May 8 '11 at 23:07
As I say, there are some obvious restrictions, but they are all obvious: If the vectors are dependent, their span has a lower dimension. If one vector is in the convex span of the others, you lose a vertex. –  Phira May 8 '11 at 23:21

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