Shellings and the lower bound theorem
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.
scientific article; zbMATH DE number 1310072
scientific article