On closed sets with convex projections under narrow sets of directions
HTML articles powered by AMS MathViewer
- by Stoyu Barov and Jan J. Dijkstra PDF
- Trans. Amer. Math. Soc. 360 (2008), 6525-6543 Request permission
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.References
- Stoyu Barov, John Cobb, and Jan J. Dijkstra, On closed sets with convex projections, J. London Math. Soc. (2) 65 (2002), no. 1, 154–166. MR 1875142, DOI 10.1112/S002461070100285X
- S. Barov and J. J. Dijkstra, On closed sets with convex projections under somewhere dense sets of projections, preprint.
- Karol Borsuk, An example of a simple arc in space whose projection in every plane has interior points, Fund. Math. 34 (1947), 272–277. MR 25721, DOI 10.4064/fm-34-1-272-277
- John Cobb, Raising dimension under all projections, Fund. Math. 144 (1994), no. 2, 119–128. MR 1273691, DOI 10.4064/fm-144-2-119-128
- Jan J. Dijkstra, Troy L. Goodsell, and David G. Wright, On compacta with convex projections, Topology Appl. 94 (1999), no. 1-3, 67–74. Special issue in memory of B. J. Ball. MR 1695348, DOI 10.1016/S0166-8641(98)00053-4
- Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna, Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221
- Branko Grünbaum, Convex polytopes, Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. MR 0226496
- Gottfried Köthe, Topologische lineare Räume. I, Die Grundlehren der mathematischen Wissenschaften, Band 107, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960 (German). MR 0130551
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
- MR Author ID: 58030
- Email: dijkstra@cs.vu.nl
- 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: https://doi.org/10.1090/S0002-9947-08-04466-8
- MathSciNet review: 2434297