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 $P$? 



The number of facets of a polyhedron under linear transformation
No. Every bounded convex polyhedron $Q$ with $v$ vertices is an image of a simplex $P$ with $v$ vertices and $v$ facets by a linear transformation. However, $Q$ may have many facets. For example, the cross-polytopes (dual to hypercubes) have $2d$ vertices and $2^d$ facets, one for each orthant, and $2^d$ is not $O(d)$. 


