# Some Characterization of Locally Nonconical Convex Sets

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## Abstract

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.

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## References

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