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Mat. Zametki, 2010, Volume 87, Issue 4, Pages 580–593 (Mi mz7698)  

This article is cited in 21 scientific papers (total in 21 papers)

On a Property of $n$-Dimensional Simplices

M. V. Nevskij

P. G. Demidov Yaroslavl State University

Abstract: Suppose that $n\in\mathbb N$ and $S$ is a simplex in $\mathbb R^n$, containing the cube $[0,1]^n$. It is proved that, for some $i=1,…,n$, the simplex $S$ contains an interval of length $n$ parallel to the $i$th coordinate axis. The relationship with questions arising in linear interpolation of continuous functions of $n$ variables is noted.

Keywords: $n$-dimensional simplex, polytope, barycentric coordinates, axial diameter, interpolation projection operator, Steiner symmetrization, Hadamard number

DOI: https://doi.org/10.4213/mzm7698

Full text: PDF file (496 kB)
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English version:
Mathematical Notes, 2010, 87:4, 543–555

Bibliographic databases:

UDC: 517.51+514.17
Received: 22.04.2009
Revised: 03.07.2009

Citation: M. V. Nevskij, “On a Property of $n$-Dimensional Simplices”, Mat. Zametki, 87:4 (2010), 580–593; Math. Notes, 87:4 (2010), 543–555

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. V. Nevskij, “On the Axial Diameters of a Convex Body”, Math. Notes, 90:2 (2011), 295–298  mathnet  crossref  crossref  mathscinet  isi
    2. M. V. Nevskii, “Geometricheskie otsenki v polinomialnoi interpolyatsii”, Model. i analiz inform. sistem, 18:1 (2011), 142–148  mathnet
    3. M. V. Nevskii, “O gipoteze Lassaka dlya vypuklogo tela”, Model. i analiz inform. sistem, 18:3 (2011), 5–11  mathnet
    4. M. V. Nevskii, “O geometricheskikh kharakteristikakh $n$-mernogo simpleksa”, Model. i analiz inform. sistem, 18:2 (2011), 52–64  mathnet
    5. Nevskii M., “Properties of Axial Diameters of a Simplex”, Discret. Comput. Geom., 46:2 (2011), 301–312  crossref  mathscinet  zmath  isi  elib  scopus
    6. M. V. Nevskii, “O nekotorykh rezultatakh po geometrii vypuklykh tel i ikh prilozheniyakh”, Model. i analiz inform. sistem, 19:3 (2012), 113–123  mathnet
    7. M. V. Nevskii, “On the Minimal Positive Homothetic Image of a Simplex Containing a Convex Body”, Math. Notes, 93:3 (2013), 470–478  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    8. M. V. Nevskii, “Ob odnoi zadache dlya simpleksa i kuba v ${\mathbb R}^n$”, Model. i analiz inform. sistem, 20:3 (2013), 77–85  mathnet
    9. M. V. Nevskii, “Computation of the longest segment of a given direction in a simplex”, J. Math. Sci., 203:6 (2014), 851–854  mathnet  crossref  mathscinet
    10. Akopyan A.V., Glazyrin A.A., “On Maximum Volume Simplices in Polytopes”, Period. Math. Hung., 69:2 (2014), 251–256  crossref  mathscinet  zmath  isi  scopus
    11. M. V. Nevskii, A. Yu. Ukhalov, “O chislovykh kharakteristikakh simpleksa i ikh otsenkakh”, Model. i analiz inform. sistem, 23:5 (2016), 603–619  mathnet  crossref  mathscinet  elib
    12. M. V. Nevskii, A. Yu. Ukhalov, “Novye otsenki chislovykh velichin, svyazannykh s simpleksom”, Model. i analiz inform. sistem, 24:1 (2017), 94–110  mathnet  crossref  mathscinet  elib
    13. M. V. Nevskii, A. Yu. Ukhalov, “On $n$-dimensional simplices satisfying inclusions $S\subset [0,1]^n\subset nS$”, Automatic Control and Computer Sciences, 52:7 (2018), 667–679  mathnet  crossref  crossref  elib
    14. M. V. Nevskii, A. Yu. Ukhalov, “O minimalnom koeffitsiente pogloscheniya dlya $n$-mernogo simpleksa”, Model. i analiz inform. sistem, 25:1 (2018), 140–150  mathnet  crossref  elib
    15. M. V. Nevskii, A. Yu. Ukhalov, “On optimal interpolation by linear functions on an $n$-dimensional cube”, Automatic Control and Computer Sciences, 52:7 (2018), 828–842  mathnet  crossref  crossref  elib
    16. M. V. Nevskii, A. Yu. Ukhalov, “Nekotorye svoistva $0/1$-simpleksov”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 18:3 (2018), 305–315  mathnet  crossref  elib
    17. Nevskii M., Ukhalov A., “Perfect Simplices in R-5”, Beitr. Algebr. Geom., 59:3 (2018), 501–521  crossref  mathscinet  zmath  isi  scopus
    18. M. V. Nevskii, “O nekotorykh zadachakh dlya simpleksa i shara v ${\mathbb R}^n$”, Model. i analiz inform. sistem, 25:6 (2018), 680–691  mathnet  crossref
    19. Nevskii M.V., Ukhalov A.Yu., “On Minimal Absorption Index For An N-Dimensional Simplex”, Autom. Control Comp. Sci., 52:7 (2018), 680–687  crossref  mathscinet  isi  scopus
    20. Nevskii M.V., Ukhalov A.Yu., “On Optimal Interpolation By Linear Functions on N-Dimensional Cube”, Autom. Control Comp. Sci., 52:7 (2018), 828–842  crossref  isi  scopus
    21. M. V. Nevskii, A. Yu. Ukhalov, “Lineinaya interpolyatsiya na evklidovom share v ${\mathbb R}^n$”, Model. i analiz inform. sistem, 26:2 (2019), 279–296  mathnet  crossref
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