The Gram-Euler theorem says that if you have a contractible polytope with straight sides, the sum of the interior solid angles at each cell (where this means the fraction of an infinitesimal sphere at that cell which is contained in the inside of the polytope), counted positively for even-dimensional cells and negatively for odd-dimensional cells, adds up to 0 (when we include all cells of dimension ≥ 0, such that the similar calculation of Euler characteristic (as though they all had solid angle 100%) would be 1), or to 1 (when we also include a unique cell of dimension -1, with angle 100%, such that the reduced Euler characteristic including this would be 0).

This is a corollary of a more specific theorem. For any direction, we may consider just those cells such that movement in this direction from the interior of that cell takes us into the interior of the polytope (we may ignore the measure zero set of directions which are parallel to some cell). These cells arrange themselves as a contractible polytope or dual polytope or something, such that the Euler characteristic of this polytope when suitably construed is 0 or 1 or whatever. Averaging this over every direction gives us our result. TODO.

There should be some connection of this to Gauss-Bonnet, thinking about angles assigned at every n-cell and the total deficit/excess as some kind of curvature. Except we have multiple dimensions to reason about now. The top dimension cell always is assigned an angle of 1, and the dimension below that is always assigned an angle of 1/2. In two dimensions, this only leaves the vertices to have nontrivial angles. In higher dimensions, we have nontrivial angles at multiple dimensions of cells. This may make the difference between having the usual Gauss-Bonnet theorem in two dimensions and not quite being able to state it easily in more dimensions in these terms. Although Wikipedia tells me there’s some analogue (Chern theorem?) in any even number of dimensions, but it doesn’t seem as straightforward as I might hope.