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Russian Mathematical Surveys, 2015, Volume 70, Issue 6, Pages 1105–1165
DOI: https://doi.org/10.1070/RM2015v070n06ABEH004975 (Mi rm9639) 		 
This article is cited in 5 scientific papers (total in 5 papers)

Virtual polytopes

G. Yu. Paninaab, I. Streinuc

a St. Petersburg State University
b St. Petersburg Institute for Informatics and Automation of the Russian Academy of Sciences
c Department of Computer Science, Smith College, Northampton, MA, USA
English version PDF (1229 kB) Citations (5) Russian version article
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DOI: https://doi.org/10.1070/RM2015v070n06ABEH004975
Abstract: Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as elements of the Grothendieck group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. The present survey connects the theory of virtual polytopes with other geometrical subjects, describes a series of geometrizations together with relations between them, and gives a selection of applications.
Bibliography: 50 titles.
Keywords: Minkowski difference, coloured polygon, polytopal function, support functions, stressed graph, McMullen's polytope algebra, Maxwell polytope.
Funding agency Grant number
Mathematisches Forschungsinstitut Oberwolfach
National Science Foundation
This work received support from Mathematisches Forschungsinstitut Oberwolfach, programmes Research-in-Pairs (2009, 2014), the National Science Foundation, Smith College, and the Euler International Mathematical Institute (St. Petersburg).
Received: 10.12.2014
Revised: 04.10.2015
Bibliographic databases:
Document Type: Article
UDC: 514.144
MSC: 52B11, 52B70, 14M25
Language: English
Original paper language: Russian
Citation: G. Yu. Panina, I. Streinu, “Virtual polytopes”, Russian Math. Surveys, 70:6 (2015), 1105–1165
Citation in format AMSBIB
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\by G.~Yu.~Panina, I.~Streinu
\paper Virtual polytopes
\jour Russian Math. Surveys
\yr 2015
\vol 70
\issue 6
\pages 1105--1165
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  • https://doi.org/10.1070/RM2015v070n06ABEH004975
  • https://www.mathnet.ru/eng/rm/v70/i6/p139
    • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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