Compact set contained in a open convex set

Question

In the finite dimensional Euclidean space, let $S$ be a compact set contained in a convex open set $C$. Then how can I find a convex compact set $S'$ such that $S \subset S' \subset C$ ? It seems far trickier than I think and extremely frustrating....

Answer 1

In Euclidean spaces the convex hull of a compact set is compact. [ Theorem 3.25 in Rudin's FA]. Hence we can take $S'$ to be the convex hull of $S$.

Answer 2

In $\mathbb R^n$, the convex hull of a closed set is closed. The convex hull of a bounded set is bounded. So the convex hull of $S$ is closed and bounded, hence compact by Bolzano-Weierstrass.

Alternatively, consider the mapping $$ \phi:\mathbb L^{n+1}\times S^n\to \mathbb R^n:((t_1,...,t_{n+1}),(x_1,...,x_{n+1}))\to\sum t_kx_k. $$ $$ \mathbb L^{n+1}=\{(t_1,...t_{n+1}):t_k\in[0,1],\sum t_k=1\} $$ The mapping is continuous. $\mathbb L^{n+1}\times S^n$ is compact. So the image of this mapping is convex and compact.