On closed sets with convex projections in Hilbert space
Volume 197 / 2007
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.
