The following open problem is taken from the book Open Problems in the Geometry and Analysis of Banach Spaces, page $40.$

Problem $84:$ Assume that $X$ is an infinite-dimensional separable Banach space such that for any pair of points $x$ and $y$ in the unit sphere of $X$ there exists a linear isometry from $X$ onto $X$ such that $Tx=y.$ Is $X$ linearly isometric to a Hilbert space?

After stating the problem, authors mentioned that the problem holds true for finite-dimensional spaces but not for non separable spaces. Then they point to the reference,page $255$ which contains the problem above.

My question: Does anyone know whether the problem is still open?If yes, what is a good reference?

The authors of the book mentioned that problem $84$ is an old classical open problem. So I suspect that there should be some papers available on solving the problem partially or fully.

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    $\begingroup$ This is a famous open problem. Google Mazur's rotation problem. $\endgroup$ – Bunyamin Sari May 18 '18 at 14:34
  • $\begingroup$ Thanks @Bunyamin Sari. Your suggestion is very helpful. By the way, may i know why is this problem famous? $\endgroup$ – Idonknow May 18 '18 at 15:39
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    $\begingroup$ Given that the Open Problem book cited at the beginning of the question is only two years old, your search for a full solution need only span the last two years of publications, preprints, conference abstracts, etc, which should make your Google search easier. It would be interesting to know if the authors of the Open Problem book will maintain a publicly available list of problems from the book that have been solved since publication. $\endgroup$ – Philip Brooker May 19 '18 at 5:42

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