Journal of Geometry

, Volume 27, Issue 1, pp 1–23 | Cite as

Sets with convex closure which are unions of two starshaped sets and families of segments which have a 2-partition

  • Marilyn Breen
Article
Received:

Abstract

The following Krasnosel'skii-type theorem is proved: Let S be a nonempty set in R2 whose closure cl S is convex and bounded. Assume that for every 9 point subset T of cl S there correspond points p1 and p2 (depending on T) such that each point of T is clearly visible via S from at least one of p1 or p2. Then S is a union of two starshaped sets. The number 9 is best possible.

Moreover, a related result yields a piercing number for families of segments in Rd: Let £ be a collection of at least 6 one-dimensional convex sets in Rd such that for every line M in Rd, at most finitely many members of £ are collinear with M. Assume that every 6 members of £ may be partitioned into two sets £1 and £2 so that ∩L ∶ L in £i ≠ φ, i = 1,2. Then £ itself has such a 2-partition. The number 6 is best possible as well.

Keywords

Related Result Point Subset 
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References

  1. [1]
    BREEN, Marilyn: ‘A Krasnosel'skii-type theorem for unions of two starshaped sets in the plane,’Pacific J. Math, to appear.Google Scholar
  2. [2]
    BREEN, Marilyn: ‘An example concerning unions of two starshaped sets in the plane,’Israel J. Math. 17(1974), 347–349.Google Scholar
  3. [3]
    BREEN, Marilyn: ‘Clear visibility and unions of two starshaped sets in the plane, ’Pacific J. Math, to appear.Google Scholar
  4. [4]
    DANZER, Ludwig and GRÜNBAUM, Branko, ‘Intersection properties of boxes in Rd,‘Combinatorica 2(3) (1982), 237–246.Google Scholar
  5. [5]
    KRASNOSEL'SKII, M.A.: ‘Sur un critère pour qu'un domain soit étoilé,‘Math. Sb. 19(61) (1946), 309–310.Google Scholar
  6. [6]
    LAWRENCE, J. F., HARE, Jr., W. R., and KENELLY, John W.: ‘Finite unions of convex sets,’Proc. Amer. Math. Soc. 34(1972), 225–228.Google Scholar
  7. [7]
    LAY, Steven R.:Convex Sets and Their Applications, John Wiley, New York, 1982.Google Scholar
  8. [8]
    VALENTINE, F.A.:Convex Sets, McGraw-Hill, New York, 1964.Google Scholar

Copyright information

© Birkhäuser Verlag 1986

Authors and Affiliations

  • Marilyn Breen
    • 1
  1. 1.Department of MathematicsUniversity of OklahomaNorman

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