Prove S is Convex: Help with Convexity

[Shaad]

How do i prove this?

Let S = {(X1, …., Xn) € R^n | Xi ≥0, X1 + … +Xn = 1}. Show that S is convex.

Suppose f(Si) = {X1, X2,..., Xn}
AND g(Si) = {X1 + X2+ ...+Xn}

but If do that, then f(Si) and g(Si), both will increase. Now I'm not sure where to go from here.

*PS: Both functions are continuous convex function of Xi (Where = 1, 2, ..., N)

Now as i mentioned earlier, both function increases, so does that also mean that S is convex?

 

[StatusX]

Your notation is confusing. What is Si, and what is the target of the function f? What are you trying to do?

Anyway, the way I'd do it is just to go back to the definition of a convex set. If X=(X1,...,Xn) and Y=(Y1,...,Yn) are in S, then show sX+(1-s)Y is in S for any s in [0,1]. This direct approach is easy enough in this situation.