Geodesic
Sets invariant under projections onto two dimensional subspaces
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 233-239.

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The Blaschke--Kakutani result characterizes inner product spaces $E$, among normed spaces of dimension at least 3, by the property that for every 2 dimensional subspace $F$ there is a norm 1 linear projection onto $F$. In this paper, we determine which closed neighborhoods $B$ of zero in a real locally convex space $E$ of dimension at least 3 have the property that for every 2 dimensional subspace $F$ there is a continuous linear projection $P$ onto $F$ with $P(B)\subseteq B$.
Classification : 46A03, 46A55, 46C05, 46C15, 52A07, 52A15
Keywords: inner product space; two dimensional subspace; projection 
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     title = {Sets invariant under projections onto  two dimensional subspaces},
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Fitzpatrick, Simon; Calvert, Bruce. Sets invariant under projections onto  two dimensional subspaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 233-239. http://geodesic.mathdoc.fr/item/CMUC_1991__32_2_a4/