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Sets invariant under projections onto two dimensional subspaces. (English) Zbl 0756.46010
In this interesting paper the authors prove: If $$B$$ is a closed neighborhood of 0 in a real locally convex topological vector space $$X$$ of dimension $$\geq 3$$. Then for all two dimensional subspaces $$M$$ there is a continuous linear projection $$P$$ of $$X$$ onto $$M$$ with $$P(B)\subset B$$, if and only if $$B$$ is the inverse image under a continuous linear map $$T$$ of:
(a) the closed unit ball in an inner product $$H$$,
(b) the closed unit ball given by a norm on $$R^ 2$$, or
(c) a closed neighborhood of 0 in $$R$$.
 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 52A15 Convex sets in $$3$$ dimensions (including convex surfaces) 46A03 General theory of locally convex spaces