The structure of the virtual polytope group relative to cylinder subgroups (Q2783044)

The author defines the virtual polytope group \({\mathcal P}^*\) to be the group generated by the (non-empty) polytopes in \(\mathbb{R}^n\) via Minkowski addition: \([P]\cdot[Q]= [P+ Q]\) (the author identifies the class of a polytope with its characteristic function). Note that \([P]^{-1}= (-1)^{\dim P}[\text{relint }P]\) in the polytope ring. Identification of polytopes by translation yields the group \(\overline{{\mathcal P}}^*\). The subgroup \(\overline{\text{Cyl}}_k\) of \(k\)-cylinders consists of the Minkowski sums of some \(n-k+1\) virtual polytopes. The author shows that there is a filtration NEWLINE\[NEWLINE\overline{{\mathcal P}}^*= \overline{\text{Cyl}}_1\supset \overline{\text{Cyl}}_2\supset\cdots\supset \overline{\text{Cyl}}_{n+ 1}= \{E\},NEWLINE\]NEWLINE with \(E\) the identity element of \(\overline{{\mathcal P}}^*\). Further, there are mutually orthogonal projectors NEWLINE\[NEWLINE\overline\delta_k: \overline{{\mathcal P}}^*\to \overline{\text{Cyl}}_k,NEWLINE\]NEWLINE whose sum is the identity; then NEWLINE\[NEWLINE\overline{\text{Cyl}}_k= \overline\delta_k \overline{{\mathcal P}}^*\oplus\cdots\oplus \overline\delta_n \overline{{\mathcal P}}^*.NEWLINE\]NEWLINE The author then goes on to discuss the relationship between scissors congruence and strong scissors congruence; in the latter, lower-dimensional components are not discarded. (These correspond to the algebra of polytopes of \textit{B. Jessen} and \textit{A. Thorup} [Math. Scand. 43, 211-240 (1978; Zbl 0398.51009)] and the polytope algebra of the reviewer [Adv. Math. 78, No. 1, 76-130 (1989; Zbl 0686.52005)], respectively; for the latter, see also [\textit{R. Morelli}, Adv. Math. 97, No. 1, 1-73 (1993; Zbl 0779.52016) and \textit{M. Brion}, Tôhoku Math. J., II. Ser. 49, No. 1, 1-32 (1997; Zbl 0881.52008)]). This has obvious implications for valuations, which are also considered.