Sum of Convex Sets in Vector Space is Convex

Theorem
Let $\Bbb F \in \set {\R, \C}$.

Let $X$ be a vector space over $\Bbb F$.

Let $A$ and $B$ be convex subsets of $X$.

Then:


 * $A + B$ is convex.

Proof
Let $x, y \in A + B$ and $\closedint 0 1$.

Then there exists $a, a' \in A$ and $b, b' \in B$ such that:


 * $x = a + b$

and:


 * $y = a' + b'$

Then:

Since $A$ and $B$ are convex, we have:


 * $t a + \paren {1 - t} a' \in A$

and:


 * $t b + \paren {1 - t} b' \in B$

so that:


 * $t x + \paren {1 - t} y \in A + B$

So $A + B$ is convex.