Geodesic
Sets invariant under projections onto one dimensional subspaces
Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 227-232.

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The Hahn--Banach theorem implies that if $m$ is a one dimensional subspace of a t.v.s. $E$, and $B$ is a circled convex body in $E$, there is a continuous linear projection $P$ onto $m$ with $P(B)\subseteq B$. We determine the sets $B$ which have the property of being invariant under projections onto lines through $0$ subject to a weak boundedness type requirement.
Classification : 46A55, 52A07, 52A10
Keywords: convex; projection; Hahn--Banach; subsets of $\Bbb R^2$ 
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     title = {Sets invariant under projections onto one  dimensional subspaces},
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Fitzpatrick, Simon; Calvert, Bruce. Sets invariant under projections onto one  dimensional subspaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 2, pp. 227-232. http://geodesic.mathdoc.fr/item/CMUC_1991__32_2_a3/