Consider a (not necessarily bounded) convex polyhedron $P\subset \mathbb{R}^n$ which has $k$ facets. Let $L:\mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.

Question1: Is there a fixed constant $C$ such that the number of facets of $L(P)$ is bounded by $Ck$ ?

Edit:

Question2: Is there any bound on the number of facets of $L(P)$ in terms of $P$?

Question3: Is there any bound on the number of (none geometrical redundant) half-spaces which defined $L(P)$ in terms of $P$?