The Geometry Junkyard


Euler's Formula, Proof 17: Valuations

This proof comes from Klain and Rota, who develop it as part of a more general theory of invariant measures on Euclidean spaces. Thanks also to Robin Chapman for helping explain it to me.

Let Ud be the vector space of functions from Rd to R, and Vd be the subspace of Ud generated by characteristic functions of compact convex sets in Rd. We define a function χd that maps Vd to real numbers, as follows:

Because D and Sum are linear operators, χ is a linear function from V to R. It can also be shown by a simple induction on dimension that, if f is the characteristic function of a compact convex set K, then χ(f) = 1: By induction, χ(fz) is the characteristic function of a closed interval, from which it is clear from the definition that sum D χ(fz) is 1. Thus also (since V has the characteristic functions of convex compact sets as its basis) χ is well defined independently of the choice of coordinates for Rd.

If g is the characteristic function of the relative interior of a polytope K, it is in Vd (by inclusion-exclusion of lower-dimensional faces) and a similar induction shows that χ(g) = (-1)m, where m is the dimension of K: Because of the independence of χ from the coordinate system, we can assume m=d. Each cross-section of K is itself a polytope, so by induction χ(gz) is (-1)m-1 times the characteristic function of an open interval, and the effect of the sum and D operators is to multiply by another factor of -1.

Now we let K be a convex polytope, and evaluate χ on its characteristic function in two different ways, by partitioning K into the disjoint union of the relative interiors of its faces. By linearity of χ, the sum of its values on these relative interiors must be the same as its value on all of K, that is,

1 = sum (-1)^m

where the sum is over the set of faces of K (including K itself). Grouping the faces by their dimensions, we get a form of Euler's formula:

1 = sum (-1)i fi

where the sum runs from i=1 to the dimension of K.


Proofs of Euler's Formula.
From the Geometry Junkyard, computational and recreational geometry pointers.
David Eppstein, Theory Group, ICS, UC Irvine.

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