Convex hulls of \(f\)- and \(\beta\)-vectors
type: Item
instance of: scholarly article
title: Convex hulls of \(f\)- and \(\beta\)-vectors
zbMATH Open document ID: 0899.52010
DOI: 10.1007/PL00009326
published in: Discrete \& Computational Geometry
publication date: 1997-01-01T00:00:00Z
review text: Simplicial complexes play an important rôle in combinatorics, not only in their own right, but as (for instance) flag complexes, order complexes of partially ordered sets, and matroid complexes. Associated with such a complex (on, say, \(n\) vertices) are the numbers \(f_j\) of its \(j\)-cells, and its Betti numbers \(\beta_j\). It is natural to ask for a complete description of the family of all corresponding \(f\)- or \(\beta\)-vectors of complexes in these classes. Less ambitiously, observe that the maxima (or minima) of linear combinations of the \(f_j\) or \(\beta_j\) are determined by the convex hulls of the \(f\)- or \(\beta\)-vectors; among such linear combinations is the Euler characteristic. In this paper, the author determines these convex hulls for each of the classes mentioned at the beginning of the review.
Mathematics Subject Classification ID: 52B70
zbMATH DE Number: 1127624
zbMATH Keywords: simplicial complex
zbMATH Keywords: face vector
zbMATH Keywords: Betti numbers
zbMATH Keywords: convex hull
author: Dmitry N. Kozlov
reviewed by: Peter McMullen
MaRDI profile type: Publication
full work available at URL: pl00009326
OpenAlex ID: W1986081965
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description: scientific article; zbMATH DE number 1127624