About Intersection of two convex polytope?

Question

the intersection of two convex hull of two polytope P and Q , is it the convex hull of the intersection of P&Q ? Conv(P) ∩ Conv(Q) = conv(P∩Q) ???.

Answer 1

Convex polytopes, Branko Grünbaum

1. If $\{K_{v}\}$ is any (finite, or infinite, denumerable or not) family of convex sets in $R^{d}$, then their intersection $\bigcap\limits_{v} K_{v}$ is also convex.