Combinatorial coefficients and the mixed volume of polytopes
type: Item
instance of: scholarly article
title: Combinatorial coefficients and the mixed volume of polytopes
zbMATH Open document ID: 0880.52005
DOI: 10.1007/BF02509508
author: O. A. Gel'fond
publication date: 1997-06-24T00:00:00Z
review text: Let \(\Delta= \Delta_1+ \cdots +\Delta_n\) be the Minkowski sum of an ordered \(n\)-tuple of convex polytopes \(\Delta_i\) in \(\mathbb{R}^n\). The \(n\)-tuple is said to be expanded if for each face \(\Gamma= \Gamma_1 +\cdots +\Gamma_n\) of \(\Delta\) among its summands \(\Gamma_i \subset \Delta_i\) there is at least one vertex. The \(n\)-tuple is said to be inductively expanded if, in addition for any \(m(0 <m <n)\), the set of the first \(m\) summands of an \(m\)-dimensional face \(\Gamma^m =\Gamma_1^m +\cdots +\Gamma^m_n\) of \(\Delta\) is expanded whenever the last \(n-m\) summands of \(\Gamma^m\) are vertices. For such special polytopes \(\Delta\) the author gives an explicit formula for the combinatorial coefficients \(k_v\) of vertices \(v\) of \(\Delta\) which means the local degree of a germ of a special continuous mapping of the boundary of \(\Delta\) upon the boundary of the positive cone (Theorem 1). Further he proves the following formula for the mixed volume of inductively expanded polytopes \[ n!V(\Delta_1, \dots, \Delta_n) =(-1)^n \sum k_v \text{det} (v_1, \dots, v_n) \] (theorem 2) conjectured earlier by the author and A. G. Khovanskii. The summation extends over all vertices \(v\in \Delta\), and \(v_i\) are the vertices of \(\Delta_i\) such that \(v=v_1 +\cdots +v_n\).
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 52B11
Mathematics Subject Classification ID: 14M25
zbMATH DE Number: 1023826
zbMATH Keywords: convex polytopes
zbMATH Keywords: combinatorial coefficients
zbMATH Keywords: mixed volume
reviewed by: Eike Hertel
MaRDI profile type: Publication
cites work: Elimination theory and Newton polytopes
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 1023826