Fiber polytopes
type: Item
instance of: scholarly article
title: Fiber polytopes
zbMATH Open document ID: 0762.52003
DOI: 10.2307/2946575
author: Bernd Sturmfels
author: Billera, Louis J.
published in: Annals of Mathematics. Second Series
publication date: 1993-01-16T00:00:00Z
review text: The authors consider projections \(\pi: P\to Q\) between convex polytopes. For each \(x\in Q\) the fiber \(\pi^{-1}(x)\) is again a convex polytope, and the average of \(\pi^{-1}(x)\) over all \(x\in Q\) is called the fiber polytope \(\Sigma(P,Q)\). Its precise definition is given in terms of the Minkowski integral which is the average of the integral over all sections of \(\pi\). It is shown that the fiber polytope can be expressed as the Minkowski sum of finitely many fibers. Throughout the paper, various interesting properties of fiber polytopes are studied, in particular an interpretation of the combinatorial structure of \(\Sigma(P,Q)\) in terms of coherent subdivisions of \(Q\), relations between the boundary complexes of \(P\) and \(Q\), special cases like zonotopes as projections of cubes and centrally symmetric polytopes as projections of cross polytopes.
reviewed by: Wolfgang Kühnel
Mathematics Subject Classification ID: 52B11
zbMATH DE Number: 91441
zbMATH Keywords: fiber polytopes
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2946575
OpenAlex ID: W4234371219
description: scientific article; zbMATH DE number 91441