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Twenty proofs of Euler's formula V - E + F - 1 = 1, which applies to convex polyhedrons, i.e., 3-dimensional polytopes, are presented at the Geometry Junkyard.

I'm interested in proofs of the more general formula for the Euler characteristic number for bounded, convex polytopes of dimension greater than three as well since the signed, refined face partition polynomials enumerating the k-dimensional faces (k=0 to n) of the n-dimensional associahedra and providing the compositional inversion of formal power series obey the extended Euler formula $$V - E + (2-D-faces) - (3-D-faces) + ... $$

$$(-1)^{n-1} ((n-1)-D-facets) + (-1)^n = 1,$$

and proofs of the formula might provide insight on derivations of the face partition polynomials. (Same applies to permutahedra and multiplicative inversion.)

I'm particularly interested in proofs related to a generalized Gauss-Bonnet theorem, proofs related to differential geometry.

1) Which of the Junkyard proofs can be extended beyond three dimensions to any n-dimensional bounded, convex polytope?

2) Do you have references to other proofs for indefinite dimensions?

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  • $\begingroup$ @SamHopkins, "indefinite" not "infinite", i.e., any finite dimension . (Although Jim Propp does address infinite according to the site.) $\endgroup$
    – Tom Copeland
    Commented Nov 14, 2019 at 1:29
  • $\begingroup$ sorry, I misread that! $\endgroup$
    – Sam Hopkins
    Commented Nov 14, 2019 at 1:30
  • $\begingroup$ Easy mistake, so I'll leave the comment just in case ... . $\endgroup$
    – Tom Copeland
    Commented Nov 14, 2019 at 1:32
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    $\begingroup$ Proof number 15, using "binary homology" (ie homology with coefficients in $\mathbb{Z}/2$) applies to any bounded convex polytope, and also works with any coefficient field. $\endgroup$
    – HJRW
    Commented Nov 14, 2019 at 14:21
  • $\begingroup$ See also " Intuitive reason why the Euler characteristic is an alternatibg sum" (math.stackexchange.com/questions/1828482/…) and "Five proofs that the Euler characteristic of a closed orientable surface is even" by Q. Yuan (qchu.wordpress.com/2014/10/14/…) $\endgroup$
    – Tom Copeland
    Commented Nov 14, 2019 at 19:28

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The most straightforward way to extend Euler's formula to all convex polytopes is to show that all convex polytopes are shellable, which was only proved in 1971 by Bruggesser & Mani. Their paper is available online here: https://www.mscand.dk/article/view/11045. This fills in a gap in an 1901 argument of Schläfli. See the paper for details.

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  • $\begingroup$ Thanks. Btw, the author of the website asked for notification of new proofs. I'm interested in tying different perspectives together (not just the "optimal" method) since this would also apply to associahedra and permutohedra and related refined face polynomials relevant to multiplicative and compositional inversion of formal series. $\endgroup$
    – Tom Copeland
    Commented Nov 14, 2019 at 1:01

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