prove that the sum and difference of two convex sets in R are convex

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- Oct 3rd 2010, 09:20 AMekbioconvex sets
prove that the sum and difference of two convex sets in R are convex

- Oct 3rd 2010, 10:30 AMDinkydoe
Just work from the definitions: If R is convex then for any $\displaystyle x,y\in R$ we have that any convex combination $\displaystyle \lambda x+(1-\lambda)y$ is element of R, with $\displaystyle \lambda\in [0,1]$

Now you need to show for 2 convex sets $\displaystyle R_1,R_2$ that,

if $\displaystyle x_1,x_2\in R_1, y_1,y_2\in R_2$ then $\displaystyle \lambda (x_1+y_1)+(1-\lambda)(x_2+y_2)$ is element of $\displaystyle R_1\oplus R_2 $

The difference is actually the same excercise...