Sum of sets in a topological space

Question

Let $A$, $B$, $C$ be subsets of $\mathbb{R}^n$ with the following properties:
$A$ is not empty and bounded and $C$ is closed and convex.
Show that:

a) If $A+B\subseteq A+C$, then $B\subseteq C$.

b) If $B$ is closed and convex with $A+B=A+C$ then, $B=C$.

Any hint or solutions, or maybe a textbook with a reference or theory usefull in this problem. Thanks in advanace.

For part $b)$ we also assume that $B$ is convex.

Answer 1

Part a is here. Part b is false. For instance, put $n=1$, $A=C=[0,1]$, $B=\{0,1\}$.