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In our lecture "Discrete Geometry 1", we are examining lineare realtions between the components of the f-vector and the h-vector of a polytope, in particular the Euler-Poincaré formula and the Dehn-Sommerville Equations. We already proved that there are no more such relations characterising a polytope.

I am working on the following exercise: Give a geometric interpretation of $$h_1(P)=f_{d-1}(P)-d$$

This is, I choose some linear function in general position $c:\mathbb{R}^d\rightarrow\mathbb{R}$ and order the vertices of $P$ according to the value under this function. The number $h_1(P)$ gives all vertices that have exactly one neighbour according to this ordering. From lecture, we know that the number $h_1(P)$ does not depend on the choice of the function $c$.

Hence, this equation tells me that the number of vertices of a polytope P, that has exactly one neighbour according to the ordering described above, is equal to the number of facets of $P$, minus the dimension $d$.

I did draw some polytops and in 2D, a triangle and a square satisfy this equation, in 3D a tetrahedron satisfies it.

But my question is: How can I give a general interpretation, for I do not see any similarities between the triangle, the square and the tetrahedron that hold for any polytops satisfying this equation.

Thank you for reading through this and thanks for any help, Martin

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