Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year

IMPACT FACTOR 2017: 0.734

CiteScore 2017: 0.70

SCImago Journal Rank (SJR) 2017: 0.695
Source Normalized Impact per Paper (SNIP) 2017: 0.891

Mathematical Citation Quotient (MCQ) 2017: 0.62

See all formats and pricing
More options …
Volume 17, Issue 4


A Krasnosel’skii-type theorem for an enlarged class of orthogonal polytopes

Marilyn Breen
Published Online: 2017-10-07 | DOI: https://doi.org/10.1515/advgeom-2017-0031


Let 𝓒 be a finite family of distinct boxes in ℝd, let S = ⋃ {C : C in 𝓒} and let G be the intersection graph of 𝓒. For each block of G, assume that the corresponding members of 𝓒 have a staircase convex union. Then S with the rectilinear metric is a median space. Moreover, if every two points of S see a common point via staircase paths in S, then S is staircase starshaped.

Keywords: Orthogonal polygon; staircase convex set; staircase starshaped set; median space

MSC 2010: Primary 52.A30; 52.A35


  • [1]

    M. Breen, An enlarged class of orthogonal polytopes having staircase convex kernels. J. Combin. Math. Combin. Comput. 101 (2017), 3–12 MR3676185 Zbl 06761134Google Scholar

  • [2]

    M. Breen, An improved Krasnosel’skiĭ-type theorem for orthogonal polygons which are starshaped via staircase paths. J. Geom. 51 (1994), 31–35. MR1298342 Zbl 0815.52005CrossrefGoogle Scholar

  • [3]

    M. Breen, Characterizing certain staircase convex sets in ℝd. Beiträge Algebra Geom. 51 (2010), 251–261. MR2650490 Zbl 1204.52008Google Scholar

  • [4]

    V. Chepoi, On staircase starshapedness in rectilinear spaces. Geom. Dedicata 63 (1996), 321–329. MR1419680 Zbl 0866.52006Google Scholar

  • [5]

    L. Danzer, B. Grünbaum, V. Klee, Helly’s theorem and its relatives. In: Proc. Sympos. Pure Math., Vol. VII, 101–180, Amer. Math. Soc. 1963. MR0157289 Zbl 0132.17401Google Scholar

  • [6]

    J. Eckhoff, Helly, Radon, and Carathéodory type theorems. In: Handbook of convex geometry, Vol. A, B, 389–448, North-Holland 1993. MR1242986 Zbl 0791.52009Google Scholar

  • [7]

    F. Harary, Graph theory. Addison-Wesley Publ. Co., Reading, Mass. 1969. MR0256911 Zbl 0182.57702Google Scholar

  • [8]

    M. Krasnosselsky, Sur un critère pour qu’un domaine soit étoilé. (Russian. French summary) Rec. Math. [Mat. Sbornik] N. S. 19(61) (1946), 309–310. MR0020248 Zbl 0061.37705Google Scholar

  • [9]

    S. R. Lay, Convex sets and their applications. Wiley-Interscience 1982. MR655598 Zbl 0492.52001Google Scholar

  • [10]

    F. A. Valentine, Convex sets. McGraw-Hill Book Co., New York-Toronto-London 1964. MR0170264 Zbl 0129.37203Google Scholar

  • [11]

    M. L. J. van de Vel, Theory of convex structures, volume 50 of North-Holland Mathematical Library. North-Holland 1993. MR1234493 Zbl 0785.52001Google Scholar

About the article

Received: 2015-09-13

Revised: 2016-02-07

Published Online: 2017-10-07

Published in Print: 2017-10-26

Citation Information: Advances in Geometry, Volume 17, Issue 4, Pages 525–532, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0031.

Export Citation

© 2017 by Walter de Gruyter Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Marilyn Breen
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2018

Comments (0)

Please log in or register to comment.
Log in