TITLE: The Minkowski sum of two convex sets is convex
QUESTION [9 upvotes]: Let $A$ and $B$ be two convex subsets in $\mathbb{R}^n$. 
  Define a set $C$ given by
  $$C = A + B = \{a + b : a \in A \mbox{ and } b \in B\}.$$
  Is $C$ a convex set?

REPLY [3 votes]: A set is convex when a convex combination of two points also belong to the set.
Then take any two points of the Minkowsi sum, which are each the sum of two points of $A$ and $B$ and form their convex sum. By linearity, they can be expressed as the sum of two convex combinations of elements of $A$ and $B$.
$$(1-t)(p_A+p_B)+t(q_A+q_B)=((1-t)p_A+tq_A)+((1-t)p_B+tq_B).$$
Then by convexity of these sets, the linear combinations are themselves in $A$ and $B$ and their sum in the Minkowski sum.