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    BAROV, STOYU and DIJKSTRA, JAN J. 2010. ON CLOSED SETS IN HILBERT SPACE WITH CONVEX PROJECTIONS UNDER SOMEWHERE DENSE SETS OF DIRECTIONS. Journal of Topology and Analysis, Vol. 02, Issue. 01, p. 123.


    Frolkina, Olga 2010. A Cantor set in <mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:msup><mml:mi mathvariant="double-struck">R</mml:mi><mml:mi>d</mml:mi></mml:msup></mml:math> with “large” projections. Topology and its Applications, Vol. 157, Issue. 4, p. 745.


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  • Journal of the London Mathematical Society, Volume 65, Issue 1
  • February 2002, pp. 154-166

ON CLOSED SETS WITH CONVEX PROJECTIONS

  • STOYU BAROV (a1) (a2), JOHN COBB (a3) and JAN J. DIJKSTRA (a1) (a4)
  • DOI: http://dx.doi.org/10.1112/S002461070100285X
  • Published online: 01 March 2002
Abstract

A shadow of a subset A of ℝn is the image of A under a projection onto a hyperplane. Let C be a closed nonconvex set in ℝn such that the closures of all its shadows are convex. If, moreover, there are n independent directions such that the closures of the shadows of C in those directions are proper subsets of the respective hyperplanes then it is shown that C contains a copy of ℝn−2. Also for every closed convex set B ‘minimal imitations’ C of B are constructed, that is, closed subsets C of B that have the same shadows as B and that are minimal with respect to dimension.

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Journal of the London Mathematical Society
  • ISSN: 0024-6107
  • EISSN: 1469-7750
  • URL: /core/journals/journal-of-the-london-mathematical-society
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