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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On closed sets with convex projections under narrow sets of directions
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by Stoyu Barov and Jan J. Dijkstra PDF
Trans. Amer. Math. Soc. 360 (2008), 6525-6543 Request permission


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:
  • Jan J. Dijkstra
  • Affiliation: Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
  • MR Author ID: 58030
  • Email:
  • 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.
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 6525-6543
  • MSC (2000): Primary 52A20, 57N15
  • DOI:
  • MathSciNet review: 2434297