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Sets invariant under projections onto one dimensional subspaces. (English) Zbl 0756.52002
The authors give the characterization of sets invariant under projections onto lines through 0 subject to a weak boundedness type requirement.
Theorem. Let $$B\neq\emptyset$$ be a closed subset of $$\mathbb{R}^ 2$$ and suppose there is $$w\in\mathbb{R}^ 2$$, $$w\neq 0$$, and $$\lambda_ n\to\infty$$ such that $$\lambda_ n^{-1}w\in B$$ or $$\lambda_ n w\not\in B$$. For every 1-dimensional subspace $$m$$, there is a linear projection $$P:\mathbb{R}^ 2\to m$$ with $$P(B)\subseteq B$$ if and only if $$B$$ is one of: (a) a subset, containing 0, of a line through 0, (b) a union of parallel lines, containing 0, (c) a bounded convex symmetric neighbourhood of 0.
Theorem. Let $$B\neq\emptyset$$ be a closed subset of a real locally convex topological vector space $$E$$ whose closed subspaces are barrelled. Suppose for all $$w$$ in a hyperplane $$W$$, there is a sequence $$\lambda_ k\to\infty$$ with $$\lambda_ k w\not\in B$$ or $$\lambda_ k^{-1}\in B$$. For all 1-dimensional subspaces $$m$$, there exists a continuous linear projection $$P: E\to m$$ such that $$P(B)\subseteq B$$ is one of: (a) a closed convex circled subset whose linear hull is closed, (b) $$S+F$$, where $$0\not\in S$$, $$S$$ is a closed subset of a 1-dimensional subspace $$\ell$$, $$S$$ is not both convex and symmetric and $$F$$ is a closed linear subspace not containing $$\ell$$.
There are several misprints of technical character in the paper [see the authors’ “Correction” in ibid. 33, No. 1, p. 182 (1992; see the following paper)].
Reviewer: J.Durdil (Praha)

##### MSC:
 52A07 Convex sets in topological vector spaces (aspects of convex geometry) 52A10 Convex sets in $$2$$ dimensions (including convex curves) 46A55 Convex sets in topological linear spaces; Choquet theory
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