Convex hull of union of nondisjoint polyhedra

Question

Is there a theorem which proves that the convex hull of the union of nondisjoint polyhedra is also polyhedral?

Answer 1

Suppose the individual polyhedra are given by $\{ x : A^i x \leq b^i\}$, the convex hull of their union is: $$\{ x : x = \sum_i \lambda_i x^i, A^i x^i \leq b^i, \sum_i \lambda_i = 1, \lambda \geq 0\}.$$ Since the projection of a polehedron onto an affine set ($x^i$ and $\lambda$ are projected out) is polyhedral, the set above is polyhedral.