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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On closed sets with convex projections under narrow sets of directions


Authors: Stoyu Barov and Jan J. Dijkstra
Journal: Trans. Amer. Math. Soc. 360 (2008), 6525-6543
MSC (2000): Primary 52A20, 57N15
Published electronically: July 28, 2008
MathSciNet review: 2434297
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Abstract: Dijkstra, Goodsell, and Wright have shown that if a nonconvex compactum in $ \mathbb{R}^n$ has the property that its projection onto all $ k$-dimensional planes is convex, then the compactum contains a topological copy of the $ (k-1)$-sphere. This theorem was extended over the class of unbounded closed sets by Barov, Cobb, and Dijkstra. We show that the results in these two papers remain valid under the much weaker assumption that the collection of projection directions has a nonempty interior.


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Additional Information

Stoyu Barov
Affiliation: Institute of Mathematics, Bulgarian Academy of Science, 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
Email: stoyu@yahoo.com

Jan J. Dijkstra
Affiliation: Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
Email: dijkstra@cs.vu.nl

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04466-8
PII: S 0002-9947(08)04466-8
Keywords: Convex projection, shadow, hyperplane, extremal point, imitation
Received by editor(s): October 20, 2004
Received by editor(s) in revised form: December 18, 2006
Published electronically: July 28, 2008
Additional Notes: The first author is pleased to thank the Vrije Universiteit Amsterdam for its hospitality and support.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.