Definition. A basis $e_1,\dots,e_n$ for a Banach space $X$ is called extremal if there exists a point $s$ in the unit sphere $S_X=\{x\in X:\|x\|=1\}$ such that for every $i\in\{1,\dots,n\}$ the intersection $(s+\mathbb R\cdot e_i)\cap B_X$ of the line $s+\mathbb R\cdot e_i$ with the unit ball $B_X=\{x\in X:\|x\|\le 1\}$ is contained in the unit sphere $S_X$.
Example 1. The standard basis $e_1,e_2$ of the Banach space $\ell_\infty(2)$ is extremal. On the other hand, the basis $e_1+e_2,e_1-e_2$ is not extremal in $\ell_\infty(2)$.
Example 2. Every basis in a smooth finite-dimensional Banach space is not extremal (a Banach space is smooth if its unit ball has a unique support hyperplane at each points of the unit sphere).
Problem. Has every finite-dimensional Banach space a non-extremal basis?
Remark. The answer to this problem is affirmative for 2-dimensional Banach spaces.
This problem was posed on 30 March 2021 by Taras Banakh on page 64 of Volume 3 of Lviv Scottish Book.
Prize. A joint non-extremal bicycle ride near Lviv with beer and pizza included.
