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
Full-text PDF Free Access
Abstract: Dijkstra, Goodsell, and Wright have shown that if a nonconvex compactum in has the property that its projection onto all -dimensional planes is convex, then the compactum contains a topological copy of the -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.
- 1. 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, https://doi.org/10.1112/S002461070100285X
- 2. S. Barov and J. J. Dijkstra, On closed sets with convex projections under somewhere dense sets of projections, preprint.
- 3. 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, https://doi.org/10.4064/fm-34-1-272-277
- 4. John Cobb, Raising dimension under all projections, Fund. Math. 144 (1994), no. 2, 119–128. MR 1273691, https://doi.org/10.4064/fm-144-2-119-128
- 5. 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, https://doi.org/10.1016/S0166-8641(98)00053-4
- 6. Ryszard Engelking, Topologia ogólna, Państwowe Wydawnictwo Naukowe, Warsaw, 1975 (Polish). Biblioteka Matematyczna. Tom 47. [Mathematics Library. Vol. 47]. MR 0500779
- 7. Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221
- 8. Branko Grünbaum, Convex polytopes, With the cooperation of Victor Klee, M. A. Perles and G. C. Shephard. Pure and Applied Mathematics, Vol. 16, Interscience Publishers John Wiley & Sons, Inc., New York, 1967. MR 0226496
- 9. Gottfried Köthe, Topologische lineare Räume. I, Die Grundlehren der mathematischen Wissenschaften, Bd. 107, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960 (German). MR 0130551
- S. Barov, J. Cobb and J. J. Dijkstra, On closed sets with convex projections, J. London Math. Soc. 65 (2002), 154-166. MR 1875142
- S. Barov and J. J. Dijkstra, On closed sets with convex projections under somewhere dense sets of projections, preprint.
- K. Borsuk, An example of a simple arc in space whose projections in every plane has interior points, Fund. Math. 34 (1947), 272-277. MR 0025721
- J. Cobb, Raising dimension under all projections, Fund. Math. 144 (1994), 119-128. MR 1273691
- J. J. Dijkstra, T. L. Goodsell, and D. G. Wright, On compacta with convex projections, Topology Appl. 94 (1999), 67-74. MR 1695348
- R. Engelking, General Topology, PWN, Warsaw, 1977. MR 0500779
- R. J. Gardner, Geometric Tomography, Cambridge Univ. Press, Cambridge, 1995. MR 1356221
- B. Grünbaum, Convex Polytopes, Pure Appl. Math. 16, Interscience, London 1967. MR 0226496
- G. Köthe, Topologische Lineare Räume I, Springer, Berlin, 1960. MR 0130551
Affiliation: Institute of Mathematics, Bulgarian Academy of Science, 8 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
Jan J. Dijkstra
Affiliation: Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands
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.