The difference between polyhedral complex and support of a polyhedral complex? A polyhedral complex is a collection of polyhedra such that intersection of any two polyhedron is a face of of both the polyhedron or empty. Support of a polyhedral complex is the set of all points in the polyhedral complex. So, what exactly is the difference between polyhedral complex and its support?

A polyhedral complex is a _set_ of polyhedra. Its support is their _union_.

In other words, a polyhedral complex has an internal structure in terms of what its constituent polyhedra are and how they are arranged. Taking the support "forgets" the internal structure and flattens it into an undifferentiated set of points.

For example, in two dimensions, you can overlay a polyhedral complex over the set $[0,1]\times[0,1]$ in many ways: as a single square-shaped polyhedron; as two isosceles right triangles; as an $n\times n$ grid of squares of side $\frac1n$; and so on. All of these are different polyhedral complexes, but they have the same support.