Czechoslovak Mathematical Journal

, Volume 54, Issue 3, pp 767–771 | Cite as

Some Characterization of Locally Nonconical Convex Sets

  • Witold Seredyński


A closed convex set Q in a local convex topological Hausdorff spaces X is called locally nonconical (LNC) if for every x, y ∈ Q there exists an open neighbourhood U of x such that \({\text{(}}U \cap Q{\text{)}} + \frac{{\text{1}}}{{\text{2}}}{\text{(}}y - x{\text{)}} \subset Q\). A set Q is local cylindric (LC) if for x, y ⊂ Q, x ≠ y, z ⊂ (x, y) there exists an open neighbourhood U of z such that U ∩ Q (equivalently: bd(Q) ∩ U) is a union of open segments parallel to [x, y]. In this paper we prove that these two notions are equivalent. The properties LNC and LC were investigated in [3], where the implication LNC ⋒ LC was proved in general, while the inverse implication was proved in case of Hilbert spaces.

stable convex set 


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  1. [1]
    J. Cel: Tietze-type theorem for locally nonconical convex sets. Bull. Soc. Roy. Sci Liège 69 (2000), 13–15.Google Scholar
  2. [2]
    S. Papadopoulou: On the geometry of stable compact convex sets. Math. Ann. 229 (1977), 193–200.Google Scholar
  3. [3]
    G. C. Shell: On the geometry of locally nonconical convex sets. Geom. Dedicata 75 (1999), 187–198.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2004

Authors and Affiliations

  • Witold Seredyński
    • 1
  1. 1.Institute of MathematicsTechnical University of WrocławWrocławPoland

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