Inclusion of polytopes

Question

Let $C_{1}$ and $C_{2}$ be polytopes in $\mathbb{R}^{n}$ such that $C_{1}=conv\left( V\right) $ with $V$ being a set of vertices. If $V\subseteq C_{2}$, my question is $C_{1}\subseteq C_{2}$?

Answer 1

• let $V$ the vertex set of a pentagon. Thus $C_1$ = pentagon.
• use for $C_2$ the vertex-inscribed pentagram, i.e. a faceting of $C_1$.
• then $C_1\nsubseteq C_2$.

--- rk

Answer 2

By definition, the convex hull $conv(V)$ is the smallest convex set containing $V$ or equivalently the intersection of all convex sets that contain $V$. Using the latter definition, if $C_2$ is convex, for any $p \in C_1$, we must have $p \in C_2$ since otherwise it wouldn’t have landed in the intersection. Therefore, $C_1 \subseteq C_2$.

As pointed out in the comments, it’s easy to find a counterexample with concave $C_2$, so you need the convexity of $C_2$.