Geodesic
A geometrical conjecture of Banach
Izvestiya. Mathematics , Tome 1 (1967) no. 5, pp. 1055-1064.

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This article is devoted to the following problem of Banach: Let $B^n$ be a Banach space of finite or infinite dimension $n$ and let $k$ be a natural number satisfying the inequalities $1$; if all the $k$-dimensional subspaces of $B^n$ are isometric to each other, is $B^n$ a Hilbert space? We give a positive answer to this question under certain restrictions on $k$ and $n$. 
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M. L. Gromov. A geometrical conjecture of Banach. Izvestiya. Mathematics , Tome 1 (1967) no. 5, pp. 1055-1064. http://geodesic.mathdoc.fr/item/IM2_1967_1_5_a7/

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