Sets invariant under projections onto one dimensional subspaces.

*(English)*Zbl 0756.52002The 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)].

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)