show that a set is convex

Question

Let $S_1, S_2 \subseteq \mathbb{R}^n$ with $S_1$ convex. Show that the following set is convex. $$\{\, x \mid x + S_2 \subseteq S_1\,\}$$

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I understand that I need to express this set as an intersection of convex sets. How should I do that?

Answer 1

Your set equals $$\bigcap_{s\in S_2}(S_1-s) $$ ans is convex as translates and intersections of convex sets are convex.

Answer 2

let $s_2\in S_2$, we have to show if $x+s_2=s_1\in S_1, y+s_2=s'_1\in S_1$, then $tx+(1-t)y+s_2\in S_1$. We have: $tx+ts_2+(1-t)y+(1-t)s_2=tx+(1-t)y+s_2=ts_1+(1-t)s_1'$.$ts_1+(1-t)s_1'\in S_1$ since $S_1$ is convex