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On Banach’s isometric subspaces problem

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Is a normed vector space V whose n-dimensional linear subspaces are all isometric, for a fixed 2 <= n < dim V, necessarily Euclidean? This question was asked in 1932 by S. Banach and in the known cases the answer is always affirmative. In a joint work with S. Ivanov and A. Nordskova we handle `the smallest’ previously unresolved case n = 3, but the problem remains open for n 1 = dim V = 4k >= 8 and n 1= dim V = 134.

I will start by formulating the problem in a couple equivalent ways, then give an overview of previous partial results, and proceed by sketching the proof in the case n = 3. If time permits, I will also discuss the local (stronger) version of the problem and its application to Finsler geometry.

This talk is part of the Junior Geometry Seminar series.

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