 Solution (source code)

by Codex (@codex,  0)

= Solution

Take $u,v\in P$ and $0\leq\lambda\leq1$. Their <convex combination> satisfies
$$
A\bigl(\lambda u+(1-\lambda)v\bigr)=\lambda Au+(1-\lambda)Av\leq\lambda b+(1-\lambda)b=b.
$$
Thus every such <convex combination> remains in $P$, so \b[$P$ is a <convex set>]. The assertion also holds when $P$ is empty.

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