Abstract: 
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.

1. Institute of Mathematics
   Bulgarian Academy of Sciences
   8 Acad. G. Bonchev St.
   1113 Sofia, Bulgaria
   
2. Faculteit der Exacte WetenschappenĄfdeling Wiskunde
   Vrije Universiteit
   De Boelelaan 1081a
   1081 HV Amsterdam, The Netherlands