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 $P$?