On closed sets with convex projections in Hilbert space

Tom 197 / 2007

Stoyu Barov, Jan J. Dijkstra Fundamenta Mathematicae 197 (2007), 17-33 MSC: 52A07, 57N20. DOI: 10.4064/fm197-0-2

Streszczenie

Let $k$ be a fixed natural number. We show that if $C$ is a closed and nonconvex set in Hilbert space such that the closures of the projections onto all $k$-hyperplanes (planes with codimension $k$) are convex and proper, then $C$ must contain a closed copy of Hilbert space. In order to prove this result we introduce for convex closed sets $B$ the set $\mathcal E^k(B)$ consisting of all points of $B$ that are extremal with respect to projections onto $k$-hyperplanes. We prove that $\mathcal E^k(B)$ is precisely the intersection of all $k$-imitations $C$ of $B$, i.e., closed sets $C$ that have the same projections as $B$ onto all $k$-hyperplanes. For every closed convex set $B$ in $\ell^2$ with nonempty interior we construct “minimal” $k$-imitations $C$, in the sense that $\mathop{\rm dim}(C\setminus\mathcal E^k(B))\le0$. Finally, we show that whenever a compact set has convex projections onto all finite-dimensional planes, then it must be convex.

Autorzy

  • Stoyu BarovInstitute of Mathematics
    Bulgarian Academy of Sciences
    8 Acad. G. Bonchev St.
    1113 Sofia, Bulgaria
    e-mail
  • Jan J. DijkstraFaculteit der Exacte Wetenschappen/Afdeling Wiskunde
    Vrije Universiteit
    De Boelelaan 1081a
    1081 HV Amsterdam, The Netherlands
    e-mail

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