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 $P(B)\subseteq B$.
@article{116961,
author = {Simon Fitzpatrick and Bruce Calvert},
title = {Sets invariant under projections onto two dimensional subspaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {32},
year = {1991},
pages = {233-239},
zbl = {0756.46010},
mrnumber = {1137784},
language = {en},
url = {http://dml.mathdoc.fr/item/116961}
}
Fitzpatrick, Simon; Calvert, Bruce. Sets invariant under projections onto two dimensional subspaces. Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) pp. 233-239. http://gdmltest.u-ga.fr/item/116961/
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