I'm reading a book on convex optimisation and understood the definition of convexity. However, I'm not able to understand the following statement.
If $S \subseteq \mathbb{R}^n$ is convex, and $a \in \mathbb{R}^n$, then the set $S+a = \{x+a \mid x \in S \}$ is convex.
How can we guarantee that every point on the line segment between $a$ and an element of $S$ will be in $S+a$?
$S + a $ is convex because if you take any two points in $S + a$, they can be written as $x + a$ and $y + a$ where $x$ and $y$ are in $S$ and then, because the line segment $[x, y]$ is contained in $S$ by the convexity of $S$, the line segment $[x + a, y + a]$ will be contained in $S + a$. $a$ will not be a member of $S + a$ in general.