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On closed sets with convex projections under somewhere dense sets of directions

Authors: Stoyu Barov and Jan J. Dijkstra
Journal: Proc. Amer. Math. Soc. 137 (2009), 2425-2435
MSC (2000): Primary 52A20, 46A55, 57N15
Published electronically: February 12, 2009
MathSciNet review: 2495278
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Abstract: Let $ k,n\in\mathbb{N}$ with $ k<n$ and let $ {\mathcal G}_k(\mathbb{R}^n)$ denote the Grassmann manifold consisting of all $ k$-dimensional linear subspaces in $ \mathbb{R}^n$. In an earlier paper the authors showed that if the projections of a nonconvex closed set $ C\subset\mathbb{R}^n$ are convex and proper for projection directions from some nonempty open set $ \mathcal{P}\subset{\mathcal G}_{k}(\mathbb{R}^n)$, then $ C$ contains a closed copy of an $ (n-k-1)$-manifold. In this paper we improve on that result by showing that that result remains valid under the weaker assumption that $ \mathcal{P}$ is somewhere dense in $ {\mathcal G}_k(\mathbb{R}^n)$.

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Additional Information

Stoyu Barov
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, 8 Acad. G. Bonchev Street, 1113 Sofia, Bulgaria

Jan J. Dijkstra
Affiliation: Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands

Keywords: Convex projection, shadow, hyperplane, imitation, Grassmann manifold, proper mapping
Received by editor(s): April 28, 2008
Received by editor(s) in revised form: October 11, 2008
Published electronically: February 12, 2009
Additional Notes: The first author is pleased to thank the Vrije Universiteit Amsterdam for its hospitality and support.
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.