"Difference" of convex set and any set

Question

Given two sets $S_1,S_2 \subset \mathbb{R}^n$ where $S_1$ is convex, is the following set always convex? $$ \{x \mid \{x + y \mid y \in S_2\} \subset S_1\} $$ I couldn't quite wrap my head around what this intuitively means. It seems like this set is some kind of "difference" between $S_1$ and $S_2$, and since $S_2$ is arbitrary I'd guess this set is not always convex... but I'm really not sure.

Answer 1

$$\{ x: x+S_2\subset S_1\} =\bigcap_{y\in S_2} (S_1 -y)$$ and hence is convex iff $S_1$ is convex.