1.3. Integral piecewise affine spaces [014Y]

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1.3. Integral piecewise affine spaces

The following discussion roughly follows [KKMS, p.59] and [Berk04, §1].

If $P$ is a rational polytope in ${\mathbb{R}}^{n}$ , that is, the convex hull of a finite subset of ${\mathbb{Q}}^{n}$ , denote by $M_{P}\subset C^{0}(P)$ the finitely generated free abelian group obtained by restricting to $P$ affine functions with coefficients in ${\mathbb{Z}}$ (constant term included). Denote by $1_{P}$ the constant function on $P$ with value $1$ , and set

\vec{M}_{P}:=M_{P}/M_{P}\cap{\mathbb{Q}}1_{P}.

Denote also by $b_{P}\in{\mathbb{N}}$ the greatest integer such that $b_{P}^{-1}1_{P}\in M_{P}$ .

The data of $(P,M_{P})$ modulo homeomorphism is called an (abstract) ${\mathbb{Z}}$ -polytope. The functions in $M_{P}$ are called integral affine, or ${\mathbb{Z}}$ -affine.

The evaluation map defines a canonical realization $P\hookrightarrow(M_{P})^{\vee}_{\mathbb{R}}$ as a codimension one rational polytope, with tangent space $T_{P}$ identified with $(\vec{M}_{P})^{\vee}_{\mathbb{R}}$ . Further, the lattice $T_{P,{\mathbb{Z}}}:=\operatorname{Hom}(\vec{M}_{P},{\mathbb{Z}})\subset T_{P}$ yields a normalized Lebesgue measure $\lambda_{P}$ on $P$ .

The main example for us is as follows.

Lemma 1.2.

Given $b_{0},\dots,b_{p}\in{\mathbb{N}}^{*}$ , view

\sigma=\left\{w\in{\mathbb{R}}_{+}^{p+1}\mid\sum_{i=0}^{p}b_{i}w_{i}=1\right\}

as a ${\mathbb{Z}}$ -simplex. Then $b_{\sigma}=\gcd(b_{i})$ , and

\operatorname{Vol}(\sigma)=\frac{b_{\sigma}}{p!\prod_{i}b_{i}}.

Proof.

Note that $T_{\sigma,{\mathbb{Z}}}=\{w\in{\mathbb{Z}}^{p+1}\mid\sum_{i}b_{i}w_{i}=0\}$ . The linear isomorphism $\phi\colon{\mathbb{R}}^{p+1}\to{\mathbb{R}}^{p+1}$ given by $\phi(w_{j})=(b_{j}w_{j})$ takes $\sigma$ to the standard simplex

\sigma^{\prime}=\{w^{\prime}\in{\mathbb{R}}_{+}^{p+1}\mid\sum_{i}w^{\prime}_{j}=1\},

and hence

[T_{\sigma^{\prime},{\mathbb{Z}}}\colon\phi(T_{\sigma,{\mathbb{Z}}})]\operatorname{Vol}(\sigma)=\operatorname{Vol}(\sigma^{\prime})=\frac{1}{p!}.

Write $T_{\sigma^{\prime},{\mathbb{Z}}}$ as the kernel of $\chi\colon{\mathbb{Z}}^{p+1}\to{\mathbb{Z}}$ defined by $\chi(w^{\prime})=\sum_{i}w^{\prime}_{i}$ . Then $\phi(T_{\sigma,{\mathbb{Z}}})=\ker\chi\cap\phi({\mathbb{Z}}^{p+1})$ , $\chi(\phi({\mathbb{Z}}^{p+1}))=\gcd(b_{i}){\mathbb{Z}}$ , and the exact sequence

0\to\frac{\ker\chi}{\ker\chi\cap\phi({\mathbb{Z}}^{p+1})}\to\frac{{\mathbb{Z}}^{p+1}}{\phi({\mathbb{Z}}^{p+1})}\to\frac{{\mathbb{Z}}}{\chi(\phi({\mathbb{Z}}^{p+1}))}\to 0

gives as desired

[T_{\sigma^{\prime},{\mathbb{Z}}}\colon\phi(T_{\sigma,{\mathbb{Z}}})]=\frac{\prod_{i}b_{i}}{\gcd(b_{i})}.

Finally, the first assertion is clear. ∎

Remark 1.3.

By setting $w_{0}=b_{0}^{-1}(1-\sum_{i=1}^{p}b_{i}w_{i})$ , we can identify $\sigma$ with the simplex $\sum_{1}^{p}b_{i}w_{i}\leq 1$ in ${\mathbb{R}}_{+}^{p}$ . The normalized Lebesgue measure on $\sigma$ is then given by $\lambda_{\sigma}=b_{\sigma}^{-1}|dw_{1}\wedge\dots\wedge dw_{p}|$ .

A compact rational polyhedron $K$ in ${\mathbb{R}}^{n}$ is a finite union of rational polytopes $P_{i}$ , which may then be arranged so that $P_{i}\cap P_{j}$ is either empty or a common face of $P_{i}$ and $P_{j}$ . We then say that $(P_{i})$ is a subdivision of $K$ , and call the subdivision simplicial if each $P_{i}$ is a simplex. A continuous function on $K$ is integral piecewise affine ( ${\mathbb{Z}}$ -PA for short) if $f|_{P_{i}}\in M_{P_{i}}$ for some subdivision of $K$ . These functions form a subgroup $\operatorname{PA}_{\mathbb{Z}}(K)\subset C^{0}(K)$ , and the data of $(K,\operatorname{PA}_{\mathbb{Z}}(K))$ modulo homeomorphism is called a compact ${\mathbb{Z}}$ -PA space.

The normalized Lebesgue measure of $K$ is defined as

\lambda_{K}=\sum_{\dim P_{i}=\dim K}{\bf 1}_{P_{i}}\lambda_{P_{i}}

for some (and hence any) subdivision into ${\mathbb{Z}}$ -polytopes.

Note that a ${\mathbb{Z}}$ -polytope $P$ can be regarded as a ${\mathbb{Z}}$ -PA space and that $M_{P}\subset\operatorname{PA}_{\mathbb{Z}}(P)$ .