Shellings and the lower bound theorem
type: Item
instance of: scholarly article
title: Shellings and the lower bound theorem
zbMATH Open document ID: 0948.52002
DOI: 10.1007/PL00009435
author: Roswitha Blind
author: G. Blind
published in: Discrete \& Computational Geometry
publication date: 2000-11-12T00:00:00Z
review text: A stacked \(d\)-polytope \(P_s\) is a simplicial polytope obtained from a simplex by successive addition of pyramids over facets. The number of \(k\)-faces \((1\leq k\leq d-1)\) of \(P_s\) is some function \(\varphi_k(n,d),\) where \(n\) is the vertex number of \(P_s.\) The lower bound theorem (LBT) asserts that for a simplicial \(d\)-polytope \(P\) with \(n\) vertices, the number \(f_k(P)\) of \(k\)-faces of \(P\) satisfies the inequality \(f_k(P)\geq\varphi_k(n,d),1\leq k\leq d-1.\) A shelling of the boundary complex \(\partial P\) of \(P\) is an ordering \(F_1,F_2,\dots,F_m\) of the facets of \(P\) so that \(F_{i+ 1}\cap\cup_{t=1}^iF_t\) is a nonempty union of \((d-2)\)-faces of \(F_{i+ 1},1\leq i\leq m-1.\) For \(1\leq i\leq m,\) let \(\mathcal K_i\) be the complex spanned by \(F_1,F_2,\dots,F_i,\) so that \(\mathcal K_m=\partial P.\) Then going from \(\mathcal K_i\) to \(\mathcal K_{i+1}\) by adding \(F_{i+1}\) is a shelling operation, in particular \(j\)-operation if \(F_{i+1}\) and \(\mathcal K_i\) have \(j\) common \((d-2)\)-faces, \(1\leq j\leq d.\) Let \(h_j\) be the number of \(j\)-operations among the \(m-1\) shelling operations. Then \(n=d+h_1,\) and \(f_1(P)=\binom d2+(d-1)h_1+h_2\) and the LBT is equivalent to the following theorem: If \(P\) is a simplicial \(d\)-polytope \((d\geq 3),\) then \(h_1\leq h_2,\) and, for \(f\geq 4,\) equality occurs only if \(P\) is stacked. This is a simple property of shellings, and the aim of this paper is to give an elementary proof of it using only shellings.
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 51M20
zbMATH DE Number: 1310072
zbMATH Keywords: lower bound theorem
zbMATH Keywords: stacked \(d\)-polytope
zbMATH Keywords: simplicial polytope
zbMATH Keywords: shelling
zbMATH Keywords: shelling operation
zbMATH Keywords: chains
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: pl00009435
OpenAlex ID: W2036824761
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description: scientific article; zbMATH DE number 1310072