Let $K,L$ be pl cell complexes and $f:|K| \to|L|$, a map that is affine linear on every cell of $K$. I need to show that $M = \{A \cap f^{-1}(B): A \in K, B\in L\}$ is a cell complex. Note that I use Rourke and Sanderson's definitions of cells, i.e. compact convex polyhedrons and faces, i.e. union of arc neighborhoods of points in a cell. I'm still struggling with the definitions so any help will be greatly appreciated.
EDIT 1. Definition of cell faces. Let $C$ be a cell in some euclidean space, $x \in C$ and let $<x,C>$ be the union of all straight lines $L$ passing through $x$ so that the intersection of each line $L$ with $C$ is a neighborhood of $x$ in $L$ (that's what I meant by arc neighborhood). Denote $<x,C> \cap C$ by $C_x$ and call it a face of $C$ with $x$ in its interior.
EDIT 2. I would like to re-open the question because I'm not sure I see it clearly. I'm trying to see if $A \cap f^{-1}(B)$ as above is a cell. It's clearly convex since $f$ is affine on $A$, but I can't see how it is a polyhedron. Can you help me? Note that Rourke and Sanderson's definition of polyhedra is that each point has a cone neighboroud fully inside the polyhedron. By cone neighborhoud of a point $x$ we mean the linear join of this point with a compact set so that the arcs from $x$ intersect only in $x$.