A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes
type: Item
instance of: scholarly article
title: A geometric approach for the upper bound theorem for Minkowski sums of convex polytopes
zbMATH Open document ID: 1365.52016
author: Menelaos I. Karavelas
author: Eleni Tzanaki
published in: Discrete \& Computational Geometry
publication date: 2016-12-20T00:00:00Z
full work available at URL: https://drops.dagstuhl.de/opus/volltexte/2015/5142/
review text: 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 in…
reviewed by: Robert Davis
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 52C45
Mathematics Subject Classification ID: 68U05
zbMATH DE Number: 6666310
zbMATH Keywords: discrete geometry
zbMATH Keywords: combinatorial geometry
zbMATH Keywords: combinatorial complexity
zbMATH Keywords: Minkowski sum
zbMATH Keywords: convex polytopes
zbMATH Keywords: upper bound
zbMATH Keywords: Cayley polytopes
describes a project that uses: OEIS
MaRDI profile type: Publication
OpenAlex ID: W1538190163
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DOI: 10.1007/S00454-016-9818-Y
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description: scientific article; zbMATH DE number 6666310