Let V be a (real) normed vector space and suppose that for a fixed 2 <= k < dim V all k-dimensional vector subspaces of V are isometric. Is it true that the norm on V is induced by an inner product?
This question was asked by Stephan Banach in 1932, and partial (affirmative) answers were given by Stanislaw Ulam et al. (1935), Arie Dvoretzky (1959), Misha Gromov (1967), and Luis Montejano et al. (2019). Recently in the preprint https://arxiv.org/abs/2204.00936
Sergei Ivanov, Anya Nordskova and myself gave a positive answer to this question in the case k = 3, but the problem remains open for k + 1 = dim V = 4l with l >= 2 and k + 1 = dim V = 134.
In the talk I will explain why the problem is natural to ask, give several reformulations of the problem, sketch the proofs in some cases, and, if time permits, give a vague idea of what goes on in our preprint. No prerequisites other than basic linear algebra will be needed to understand most of the talk.