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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 6(426), Pages 139–202 (Mi umn9639)  

This article is cited in 2 scientific papers (total in 2 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

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).


DOI: https://doi.org/10.4213/rm9639

Full text: PDF file (1312 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2015, 70:6, 1105–1165

Bibliographic databases:

Document Type: Article
UDC: 514.144
MSC: 52B11, 52B70, 14M25
Received: 10.12.2014
Revised: 04.10.2015

Citation: G. Yu. Panina, I. Streinu, “Virtual polytopes”, Uspekhi Mat. Nauk, 70:6(426) (2015), 139–202; Russian Math. Surveys, 70:6 (2015), 1105–1165

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Nekrasov I., Panina G., Zhukova A., “Cyclopermutohedron: Geometry and Topology”, Eur. J. Math., 2:3 (2016), 835–852  crossref  mathscinet  zmath  isi  scopus
    2. Schneider R., “The Typical Irregularity of Virtual Convex Bodies”, J. Convex Anal., 25:1 (2018), 103–118  mathscinet  zmath  isi
  • Успехи математических наук Russian Mathematical Surveys
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