Prove a theorem about a vector space and convex sets

[LCSphysicist]

Summary:: Be the set X of vectors {x1,...,xn} belong to the vector space E. If this set X is convex, prove that all the convex combination of X yet belong to X. Where convex combination are the expression t1*x1 + t2*x2 + ... + tn*xn where t1,...,tn >= 0 and t1 + ... + tn = 1

I tried to suppose xn > xn-1 > ... > x1, so in this way we have two limits, and the convex requires that all elements of E [v1,vn] belongs to X.
Now here i smell a rat: I suppose that xn > t1*x1 + t2*x2 + ... + tn*xn > x1, in such way that it will automatically belongs to E. The problem is how to prove my statement...

[Moderator's note: Moved from a technical forum and thus no template.]

 

[Office_Shredder]

xn > t1*x1 + t2*x2 + ... + tn*xn > x1

Remember these are vectors, you don't have a natural way to compare them. Convexity of E only says this sum is in E if it happens to lie on the line between x1 and xn, which is very unlikely for random coefficients.

I would suggest trying to do induction. n=2 is as easy as you think. What about n=3? This is just a triangle, so a picture might help you think about it.