Poincare-Hopf theorem for polytopes? Is there an analogue of Poincare-Hopf theorem for polytopes?
I want to apply it in the following situation.
I have a polytope in $R^n$ and a smooth explicitly given vector field in $R^n$.
I want to find the number of points where the field is normal to the surface of the polytope (rather, lies in the normal cone). If the polytope was a smooth compact manifold, I'd project the field to the tangent space and apply Poincare-Hopf theorem. Is there a version of Poincare-Hopf theorem which lets me do something similar for polytopes?
 A: There are certainly analogous theorems but the most direct analogy you might not find particularly useful.  For example, manifolds with boundary could be considered a step towards a general Poincare-Hopf for polyhedra.  
The most direct version of Poincare-Hopf for manifolds with boundary requires that the vector field be outward-pointing (or inward-pointing) along the boundary -- you do have to be consistent, though, i.e. outward pointing at all boundary facets, or inward pointing on all boundary facets.   The result is the same.    So it is not enough here to state on the boundary facets that your vector field is simply "normal".  This is where the biggest constraint comes in, attempting to generalize to polytopes.
Euler characteristic is "additive", so you could use the above as a guide towards attempting to generalize the theorem. 
Consider for example a polyhedron which is a Y-graph.  The euler characteristic is 1.  But if you demand a vector field to be "normal" on the three "boundary" vertices, the simplest vector field you can imagine has just two zeros, both with the same degree.  
I think the simplest thing to do in your case is to use the fact that your polytope is sitting in Euclidean space.  So take a small regular neighbourhood of it.  Provided your vector field has finitely-many zeros and is outward-pointing on the boundary, the regular Poincare-Hopf theorem applies.  Since your regular neighbourhood has the polytope as a deformation-retract they have the same Euler characteristic, so you have the generalization you desire. 
Ensuring the vector field is outward-pointing on the boundary is the only real technical condition for this theorem.  You can state this as a kind of transversality condition for the vector field on the polytope.  If this is the kind of thing you think would suffice, I could elaborate on this. 
