1

How can I prove this? If $A, B ⊂ R^L$ are convex sets, then $A + B$ is also convex, where

$A + B =\{c ∈ R^L :$ There is $a ∈ A , b ∈ B$ such that $c=a+b\}$

Bernard
  • 179,256
user707254
  • 43

1 Answers1

2

Let $c,c'\in A+B$. Let $a,a'\in A$ and $b,b'\in B$ such that $c=a+b$ and $c'=a'+b'$.

If $\lambda\in[0,1]$, $\lambda c+(1-\lambda)c'=(\lambda a+(1-\lambda)a')+(\lambda b+(1-\lambda)b')$. Because $A$ is convex, $\lambda a+(1-\lambda)a'\in A$ and because $B$ is convex, $\lambda b+(1-\lambda)b'\in B$.

Therefore, $\lambda c+(1-\lambda)c'\in A+B$ which proves that $A+B$ is convex.

Victor Colomb
  • 183