partitions and a characterization for compact unions of starshaped sets
Author:
Marilyn Breen
Journal:
Proc. Amer. Math. Soc. 102 (1988), 677680
MSC:
Primary 52A30; Secondary 52A35
MathSciNet review:
929001
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Abstract: For natural numbers and , define , and otherwise. The set in has property if and only if is a finite union of onedimensional convex sets and for every member subset of there correspond points (depending on ) such that each point of sees via some . The following results are established. (1) Let and be fixed natural numbers, and let be a collection of sets such that every members meet in at most one point. Then has a partition with if and only if every or fewer members of have such a partition. (2) Let be compact in . The set is a union of starshaped sets if and only if there is a sequence of compact sets converging to (relative to the Hausdorff metric) such that each set satisfies property . The first result yields a piercing number for certain families of sets, while the second provides a characterization for compact unions of starshaped sets in .
 [1]
Marilyn
Breen, A characterization theorem for compact unions of two
starshaped sets in 𝑅³, Pacific J. Math.
128 (1987), no. 1, 63–72. MR 883377
(88e:52011)
 [2]
Marilyn
Breen, A Krasnosel′skiĭtype theorem for unions of two
starshaped sets in the plane, Pacific J. Math. 120
(1985), no. 1, 19–31. MR 808926
(87c:52014)
 [3]
Marilyn
Breen, Clear visibility and unions of two starshaped sets in the
plane, Pacific J. Math. 115 (1984), no. 2,
267–275. MR
765187 (86b:52003)
 [4]
Marilyn
Breen, Sets with convex closure which are unions of two starshaped
sets and families of segments which have a 2partition, J. Geom.
27 (1986), no. 1, 1–23. MR 858238
(88a:52008), http://dx.doi.org/10.1007/BF01230330
 [5]
Ludwig
Danzer and Branko
Grünbaum, Intersection properties of boxes in
𝑅^{𝑑}, Combinatorica 2 (1982),
no. 3, 237–246. MR 698651
(84g:52014), http://dx.doi.org/10.1007/BF02579232
 [6]
Ludwig Danzer, Branko Grünbaum, and Victor Klee, Helly's theorem and its relatives, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R.I., 1963, pp. 101180.
 [7]
M.
Krasnosselsky, Sur un critère pour qu’un domaine soit
étoilé, Rec. Math. [Mat. Sbornik] N. S.
19(61) (1946), 309–310 (Russian, with French
summary). MR
0020248 (8,525a)
 [8]
Steven
R. Lay, Convex sets and their applications, John Wiley &
Sons, Inc., New York, 1982. Pure and Applied Mathematics; A
WileyInterscience Publication. MR 655598
(83e:52001)
 [9]
Sam
B. Nadler Jr., Hyperspaces of sets, Marcel Dekker, Inc., New
YorkBasel, 1978. A text with research questions; Monographs and Textbooks
in Pure and Applied Mathematics, Vol. 49. MR 0500811
(58 #18330)
 [10]
Frederick
A. Valentine, Convex sets, McGrawHill Series in Higher
Mathematics, McGrawHill Book Co., New YorkTorontoLondon, 1964. MR 0170264
(30 #503)
 [1]
 Marilyn Breen, A characterization theorem for compact unions of two starshaped sets in , Pacific J. Math. 128 (1987), 6372. MR 883377 (88e:52011)
 [2]
 , A Krasnosel'skiitype theorem for unions of two starshaped sets in the plane, Pacific J. Math. 120 (1985), 1931. MR 808926 (87c:52014)
 [3]
 , Clear visibility and unions of two starshaped sets in the plane, Pacific J. Math. 115 (1984), 267275. MR 765187 (86b:52003)
 [4]
 , Sets with convex closure which are a union of two starshaped sets and families of segments which have a partition, J. Geom. 27 (1986), 123. MR 858238 (88a:52008)
 [5]
 Ludwig Danzer and Branko Grünbaum, Intersection properties of boxes in , Combinatorica 2 (1982), 237246. MR 698651 (84g:52014)
 [6]
 Ludwig Danzer, Branko Grünbaum, and Victor Klee, Helly's theorem and its relatives, Proc. Sympos. Pure Math., vol. 7, Amer. Math. Soc., Providence, R.I., 1963, pp. 101180.
 [7]
 M. A. Krasnosel'skii, Sur un critère pour qu'un domaine soit étoilé, Math. Sb. 19 (61) (1946), 309310. MR 0020248 (8:525a)
 [8]
 Steven R. Lay, Convex sets and their applications, Wiley, New York, 1982. MR 655598 (83e:52001)
 [9]
 S. Nadler, Hyperspaces of sets, Dekker, New York, 1978. MR 0500811 (58:18330)
 [10]
 F. A. Valentine, Convex sets, McGrawHill, New York, 1964. MR 0170264 (30:503)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198809290016
PII:
S 00029939(1988)09290016
Keywords:
partitions,
unions of starshaped sets,
Krasnosel'skiitype theorems
Article copyright:
© Copyright 1988
American Mathematical Society
