Title: Stability in the Banach isometric conjecture for planar sections
Abstract: Banach asked whether a normed space all whose k-dimensional linear subspaces are isometric to each other, for some fixed 2 \leq k < \dim (V), must necessarily be Euclidean. At present, an affirmative answer is known for k=2 (Auerbach-Mazur-Ulam, 1935), all even k (Gromov, 1967), all k=1 \mod 4 but k=133 (Bor-Hernandez Lamoneda-Jimenez Desantiago-Montejano Peimbert, 2021), and k=3 (Ivanov-Mamaev-Nordskova, 2023). These developments, except perhaps the recent resolution of the k=3 case, can be considered spiritual successors to the original argument of Auerbach-Mazur-Ulam for k=2 which is based on a topological obstruction. In this talk, I will present a stable version of their result: if all 2-dimensional linear subspaces are approximately isometric, then the normed space is approximately Euclidean. Based on joint work with Dmitry Faifman (TAU).