# A question about pyramids (polytopes)

Let's use the following definition of a face:

A nonempty convex subset $F$ of a convex set $C$ is called a face of $C$ if $\alpha x + (1-\alpha) y \in F$ with $x, y \in C$ and $0 < \alpha < 1$ imply $x, y \in F$.

Let's focus on polytopes. In this context, say that a face $F$ of a $d$-polytope $P$ is a facet if F is a $(d-1)$-face. Moreover, say that a $d$-polytope $P$ is a pyramid with base $B$ if there is a $(d-1)$-polytope $B$ such that $P = \text{conv}(B \cup \{a\})$ for some point $a$.

My question is the following: Let $P$ be a pyramid with the above notation. Let $\widehat B$ be a facet of the base $B$. How can I prove that $\text{conv}(\widehat B \cup \{a\})$ is a facet of $P$?

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