Anya Nordskova (Hasselt) will talk in the differential geometry seminar at 2pm on Monday 9th October. The talk will be in the Salle des Profs (9th floor, building NO, Campus de la Plaine). Anya’s title is “Banach’s problem in dimension 4” and her abstract is below.
90 years ago S. Banach asked the following question. Let V be a normed vector space and assume that all its subspaces of a fixed finite dimension k, where 1 < k < n = dim V, are isometric to each other. Is B necessarily Euclidean (that is, the norm is induced by an inner product)? Translating the question into the language of convex sets: Let B be a convex centrally symmetric body in an n-dimensional normed space and assume that all its cross-sections by k-dimensional vector subspaces are linearly equivalent to each other. Is B necessarily an ellipsoid?
In general, the question remains open, but affirmative answers were given in many special cases by Auerbach, Mazur and Ulam (1935), Dvoretzky (1959), Gromov (1967), Milman (1971), Bor, Hernandez-Lamoneda, Jimenez-Desantiago (2019). Almost all of these works are based on methods of algebraic topology.
Together with S. Ivanov and D. Mamaev we managed to solve Banach’s problem in the smallest previously unknown case, for k+1=n= 4. Due to the parallelizability of the three-dimensional sphere, topological arguments used in previous works do not provide any information in our case. Hence, we develop a different, differential geometric approach.
In the talk I will give several reformulations of Banach’s problem, overview existing results and approaches, and try to sketch our proof.