Definition. A basis $e_1,\dots,e_n$ of a finite-dimensional Banach space $X$ is called Mazur-Ulam if all vectors $e_1,\dots,e_n$ have norm one and every self-isometry $f:S_X\to S_X$ of the unit sphere $S_X=\{x\in X:\|x\|=1\}$ that does not move the basic vectors is the identity map of $S_X$. Banach spaces are considered over the field of real numbers.
Problem 1. Has every finite-dimensional Banach space a Mazur-Ulam basis?
Problem 2. Is every basis in a finite-dimensional Banach space a Mazur-Ulam basis?
This problem was posed on 30 March 2021 by Taras Banakh on page 63 of Volume 3 of Lviv Scottish Book.
Prize. A ticket to Lviv Opera Theater.
Remark 1. The answer to Problem 2 (resp. 1) is affirmative for Banach spaces of dimension $\le 2$ (resp. $\le 3$).
Remark 2. This problem is a partial case of a more general (still unsolved) Tingley's problem asking whether every bijective isometry between the unit spheres of two Banach spaces extends to a linear isometry of the Banach spaces.
