Below we have the definition of a convex set.I want to prove that sum of convex sets is a convex set.using definition bellow i take two points from each set $x'_1,x'_2\in S1$ and $x''_1, x''_2 \in S2$. For each set we have the following expression
$$\lambda'x'_1+(1-\lambda')x'_2 \in S1 \\
\lambda''x''_1+(1-\lambda'')x''_2 \in S2 $$
If i want to show that their sum is convex too,i need to arrange it in the $\lambda x+(1-\lambda)x \in S1$ structure structure.
How can i do it?
Thanks
Update:I have seen solutions to these proof as shown bellow.
the main problem with this proof is that they share the same lambda unlike what i tried to have separated lambdas.
Why is that?
Best Answer
Given: $S_1$ and $S_2$ are convex sets.
To Show: $S_1+S_2$ is also convex. Note that $S_1+S_2=\{s;s=s_1+s_2 \text{ such that } s_1\in S_1\text{and }s_2\in S_2\}$
Let $s,t\in S_1+S_2$. Fix some $\lambda\in[0,1]$. (I guess here is where the doubt is. As we have to show convexity of the set $S_1+S_2$, we need not see them as separate entities, we only need to keep in mind the form of the components in that set).
Then $s=s_1+s_2, t=t_1+t_2$ such that $s_1,t_1\in S_1$ and $s_2,t_2\in S_2$.
$\lambda s+(1-\lambda)t=\lambda(s_1+s_2)+(1-\lambda)(t_1+t_2)=(\lambda s_1+(1-\lambda)t_1)+(\lambda s_2+(1-\lambda)t_2)\in S_1+S_2$. (As both sets $S_1$ and $S_2$ are convex). Thus, $S_1+S_2$ is also convex.
Hope this helps.