Banach conjecture

There was a paper today in the arXiv mailing list (arXiv:2006.00336) proving yet another case of the Banach conjecture. I never heard of this conjecture before, but it is easy to state and seems to me to be a foundational recognition principle for those Banach spaces that are actually Hilbert spaces.

The conjecture was stated by Stefan Banach in 1932 and reads as follows: if \(V\) is a Banach space (it may be real or complex, and of finite or infinite dimension) such that for some natural number \(n\) with \(2 \le n < \mathrm{dim}(V)\) all \(n\)-dimensional subspaces of \(V\) are pairwise isometrically isomorphic to each other, then \(V\) is a Hilbert space.

Many cases of the Banach conjecture are already known (for example, the \(\infty\)-dimensional case in both the real and the complex version). The above cited arXiv preprint summarizes all the single results in its introduction – so have a look if you are interested.