We have this theorem:
Let $P = conv(V) \subset \mathbb{R}^d $ be a d-polytope. Let B be the Gale transform of $\{v_1,...,v_n\}$. Then $conv\{v_j|j \in J\}$ is a face of P iff either $J=[n]$ or $0 \in relint(conv\{b_k|k \not\in J\}) \subset \mathbb{R}^{n-d-1} $
When it comes to actual problems that I want to solve by hand, such as these two:
the Gale diagram (for the first problem) makes it hard to read off the facets.
Are there any other ways to do it by hand?