Let S be a nonempty subset of a linear space, and let m dim S dim aff S Suppose that x belongs to conv S so that there exist x1, x2, , xn S and a1, a2, , an R with a1, , a2 an 1 such that x a1x1 a2x2 anxn 10 1 If n dim S 1, show that the elements x1, x2, , xn S are affinely dependent, and therefore...

1 Let x 1 x 1 2 x 2 x 121 If dim 1 the elements x 1 x 2 x are affinel ...

Let S be a nonempty subset of a linear space, and let m = dim S =

Question:

Let S be a nonempty subset of a linear space, and let m = dim S = dim aff S. Suppose that x belongs to conv S so that there exist x1, x2, . . . , xn ˆˆ S and a1, a2, . . . , an ˆˆ R+ with a1,......, a2+...........+ an = 1 such that
x = a1x1 + a2x2 +..............+ anxn … 10†
1. If n > dim S + 1, show that the elements x1, x2, . . . , xn ˆˆ S are affinely dependent, and therefore there exist numbers b1, b2, . . . , bn, not all zero, such that

And
Î²1 + Î²2 +...............+ Î²n = 0
2. Show that for any number t, x can be represented as

3. Let t = mini{ai/Î²i: Î²i > 0}. Show that ai - tÎ²i > 0 for every t and ai - tÎ²i = 0 for at least one t. For this particular t, (12) is a convex representation of x using only n - 1 elements.
4. Conclude that every x ˆˆ conv S can be expressed as a convex combination of at most dim S + 1 elements.