# 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 $$U^d$$ be the vector space of functions from $$\mathbb{R}^d$$ to $$\mathbb{R}$$, and $$V^d$$ be the subspace of $$U^d$$ generated by characteristic functions of compact convex sets in $$\mathbb{R}^d$$. We define a function $$\chi_d$$ that maps $$V^d$$ to real numbers, as follows:

• If $$d=1$$, for any function $$f$$ in $$V^d$$, let $\operatorname{D} f(x) = f(x) - \lim_{y\to x^+} f(y)$ $\chi_1(f) = \sum_{x\in\mathbb{R}} D f(x)$ (Note that this sum always has only a finite number of nonzero terms.)

• If $$d\gt 1$$, partition the coordinates of $$\mathbb{R}^d$$ into sets of $$d-1$$ and $$1$$ coordinate: $\mathbb{R}^d = \mathbb{R}^{d-1}\times\mathbb{R}.$ Thus we can represent any point in $$\mathbb{R}^d$$ as a pair $$(x,z)$$ where $$x$$ is in $$\mathbb{R}^{d-1}$$ and $$z$$ is in $$\mathbb{R}$$. For any function $$f$$ in $$V^d$$, and any $$z$$ in $$\mathbb{R}$$, let $$f_z$$ denote the restriction of $$f$$ to a $$(d-1)$$-dimensional cross-section: $$f_z(x)=f(x,z)$$, so $$f_z$$ is a function in $$V^{d-1}$$. We then define $\chi_d(f) = \sum_{z\in\mathbb{R}} \operatorname{D} \chi_{d-1}(f_z).$

Because $$\operatorname{D}$$ and $$\sum$$ are linear operators, $$\chi$$ is a linear function from $$V^d$$ to $$\mathbb{R}$$. 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 $$\chi(f)=1$$: By induction, $$\chi(f_z)$$ is the characteristic function of a closed interval, from which it is clear from the definition that $$\sum\operatorname{D}\chi(f_z)$$ is $$1$$. Thus also (since $$V^d$$ has the characteristic functions of convex compact sets as its basis) $$\chi$$ is well defined independently of the choice of coordinates for $$\mathbb{R}^d$$.

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

Now we let $$K$$ be a convex polytope, and evaluate $$\chi$$ 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 $$\chi$$, 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: $1 = \sum_{i=0}^d (-1)^i f_i.$

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