Let $f:\mathbb{R}^d\to\mathbb{R}^d$ satisfy the property that for any polyhedron $P = \{x\in\mathbb{R}^d : Ax\le b\}$, $f(P)$ is also a polyhedron. Are there known characterizations of function that satisfy this property? Must such an $f$ be a linear map? My guess is that $f$ must be piecewise linear, that is, there exist halfspaces partitioning $\mathbb{R}^d$ such that $f$ is linear when restricted to any connected component formed by these halfspaces. Any intuition/suggestions towards proving such a statement would be much appreciated!
(This question is essentially asking for a converse of the obvious statement that if $f$ is linear, then $f(P)$ is a polyhedron for any polyhedron $P$).