Are two halves of a convex polytope themselves convex?

Question

Suppose I have an $n$-dimensional convex polytope $P$ which is the convex hull of some set of vertices $V$. Now suppose I take a hyperplane $T$ that intersects $P$ and slice $P$ into two parts: $P_1$, which is to one side of the hyperplane, and $P_2$, which is on the other. It seems intuitive to me that $P_1$ and $P_2$ should both be convex. Is this true in general? And is there a theorem that I could cite when making this assertion?

Answer 1

The intersection of two convex sets (or infinitely many convex sets, for that matter) is convex. That's not hard to prove by using the definition of convexity.

And half-spaces are convex.