Hi there

How would I go about proving that "S= (x1,x2): x1 + x2 ≤ 1 , x1 ≥ 0)" is a convex set?

Thanks

*Sorry for the title typo

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- May 20th 2010, 04:40 AMLior539Proving a noSet is Convex
Hi there

How would I go about proving that "S= (x1,x2): x1 + x2 ≤ 1 , x1 ≥ 0)" is a convex set?

Thanks

*Sorry for the title typo - May 20th 2010, 06:50 AMtonio

Take any two points $\displaystyle u:=(x_1,x_2),\,v:=(y_1,y_2)\in S$ ; you must prove that $\displaystyle tu+(1-t)v\in S\,,\,\,\forall\,t\in [0,1]$:

$\displaystyle tu+(1-t)v=\left(tx_1+(1-t)y_1,\,tx_2+(1-t)y_2\right)$ , and now you have the simple task yo show that:

1) $\displaystyle tx_1+(1-t)y_1\geq 0$ ;

2) $\displaystyle tx_1+(1-t)y_1+tx_2++(1-t)y_2\geq 1$ ...(Happy)

Tonio - May 20th 2010, 07:31 AMLior539
Thanks for your help Tonio, it really cleared up some things in my mind. How would one go about proving (1) and (2) though?

Thanks :)