A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes
In [Publ. Math., Inst. Hautes Étud. Sci. 124, 99--163 (2016; Zbl 1368.52016)], \textit{K. A. Adiprasito} and \textit{R. Sanyal} stated and proved an upper bound theorem for the number of faces of Minkowski sums of polytopes (UBTM). Their approach relied on developing a relative version of Stanley-Reisner theory and computing homology. In contrast, the article under review derives the UBTM from a completely geometric perspective, bypassing the need for homological methods. This article begins by outlining McMullen's proof of the Upper Bound Theorem (UBT) for polytopes and detailing which steps of theirs differ from the steps taken by Adiprasito and Sanyal. In the second section, background about polytopes and polytopal complexes are given; particular attention is given to discussing the relationship between Minkowski sums and Cayley polytopes. If \(P_1,\dots,P_r\) are polytopes, then their Cayley polytope \(\mathcal{C}_{[r]}\) may not be simplicial. In Section 3, the authors instead construct a simplicial polytope \(\mathcal{Q}_{[r]}\) from \(\mathcal{C}_{[r]}\) via stellar subdivision, and Section 4 demonstrates that an analogue of the Dehn-Sommerville relations holds for certain subcomplex of the boundary complex for \(\mathcal{C}_R\), the Cayley polytope of the \(P_i\) for \(i \in R\) and \(R \subseteq [r]\). Namely, the authors consider the subcomplex \(\mathcal{F}_R\), consisting of all faces of \(\partial \mathcal{C}_R\) containing at least one vertex of some \(P_i\) for \(i \in R\). Section 5 establishes an upper bound for the \(h_{k+1}(\mathcal{F}_R)\) when \(0 \leq k \leq d + |R| - 2\) via a recurrence relation. This requires multiple nontrivial steps; perhaps most crucial step is in providing an upper bound on \(g\)-vectors via a particular shelling of \(\partial \mathcal{Q}_R\). To do this, the authors use a correspondence between \(h\)-numbers and an orientation of the dual graph of a simplicial polytope. This setup allows the authors to prove the main inequality of the section by generalizing part of McMullen's proof of the UBT. The following sections derive the main results of the article. Section 6 establishes the upper bounds in the UBTM in much the same way as Adiprasito and Sanyal (acknowledged by the authors themselves). Finally, Section 7 gives tightness on the upper bounds. Several appendices follow, filling in portions of proofs that had been delayed in order to keep a pleasant flow to the paper.
scientific article; zbMATH DE number 6666310
scientific article