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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?

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Can you think of two different polyhedral complexes which have the same support? –  Rahul Apr 1 '12 at 20:54
    
Well, I don't understand what exactly is the difference. If two polyhedral complexes have the same support, shouldn't they be the same polyhedral complex, according to the definition of polyhedral complex? –  Mohan Apr 2 '12 at 7:40

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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.

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